stan::math Namespace Reference

Detailed Description

Matrices and templated mathematical functions.

Templated probability distributions. All paramaterizations are based on Bayesian Data Analysis. Function gradients via reverse-mode automatic differentiation.

Namespaces
	internal
	opencl
	opencl_kernels

Classes
class	accumulator
	Class to accumulate values and eventually return their sum. More...
struct	acos_fun
	Structure to wrap acos() so it can be vectorized. More...
struct	acosh_fun
	Structure to wrap acosh() so it can be vectorized. More...
struct	ad_promotable
	If the type From can be converted to To using implicit conversions, or both From and To are possibly cv-qualified void), provides the member constant value equal to true. More...
class	ad_tape_observer
	TBB observer object which is a callback hook called whenever the TBB scheduler adds a new thread to the TBB managed threadpool. More...
class	addition_
	Represents addition in kernel generator expressions. More...
struct	adj_jac_vari
	adj_jac_vari interfaces a user supplied functor with the reverse mode autodiff. More...
struct	algebra_solver_vari
	The vari class for the algebraic solver. More...
struct	append_return_type
	This template metaprogram is used to compute the return type for append_array. More...
struct	append_return_type< Eigen::Matrix< T1, R, C >, Eigen::Matrix< T2, R, C > >
	This template metaprogram is used to compute the return type for append_array.
struct	append_return_type< int, int >
	This template metaprogram is used to compute the return type for append_array. More...
struct	append_return_type< std::vector< T1 >, std::vector< T2 > >
	This template metaprogram is used to compute the return type for append_array.
struct	apply_scalar_unary
	Base template class for vectorization of unary scalar functions defined by a template class F to a scalar, standard library vector, or Eigen dense matrix expression template. More...
struct	apply_scalar_unary< F, double >
	Template specialization for vectorized functions applying to double arguments. More...
struct	apply_scalar_unary< F, fvar< T > >
	Template specialization to fvar for vectorizing a unary scalar function. More...
struct	apply_scalar_unary< F, int >
	Template specialization for vectorized functions applying to integer arguments. More...
struct	apply_scalar_unary< F, std::vector< T > >
	Template specialization for vectorized functions applying to standard vector containers.
struct	apply_scalar_unary< F, T, require_eigen_t< T > >
	Template specialization for vectorized functions applying to Eigen matrix arguments. More...
struct	apply_scalar_unary< F, var >
	Template specialization to var for vectorizing a unary scalar function. More...
class	array_builder
	Structure for building up arrays in an expression (rather than in statements) using an argumentchaining add() method and a getter method array() to return the result. More...
struct	asin_fun
	Structure to wrap asin() so it can be vectorized. More...
struct	asinh_fun
	Structure to wrap asinh() so it can be vectorized. More...
struct	atan_fun
	Structure to wrap atan() so it can be vectorized. More...
struct	atanh_fun
	Structure to wrap atanh() so it can be vectorized. More...
struct	AutodiffStackSingleton
	This struct always provides access to the autodiff stack using the singleton pattern. More...
class	binary_operation
	Represents a binary operation in kernel generator expressions. More...
struct	cbrt_fun
	Structure to wrap cbrt() so it can be vectorized. More...
struct	ceil_fun
	Structure to wrap ceil() so it can be vectorized. More...
class	chainable_alloc
	A chainable_alloc is an object which is constructed and destructed normally but the memory lifespan is managed along with the arena allocator for the gradient calculation. More...
struct	child_type
	Primary template class for metaprogram to compute child type of T. More...
struct	child_type< T_struct< T_child > >
	Specialization for template classes / structs. More...
class	cholesky_block
class	cholesky_opencl
class	cholesky_scalar
struct	common_type
	Struct which calculates type promotion given two types. More...
struct	common_type< Eigen::Matrix< T1, R, C >, Eigen::Matrix< T2, R, C > >
	Struct which calculates type promotion over two types.
struct	common_type< std::vector< T1 >, std::vector< T2 > >
	Struct which calculates type promotion over two types.
struct	conjunction
	Extends std::true_type when instantiated with zero or more template parameters, all of which extend the std::true_type. More...
struct	conjunction< T, Ts... >
struct	cos_fun
	Structure to wrap cos() so it can be vectorized. More...
struct	cosh_fun
	Structure to wrap cosh() so it can be vectorized. More...
struct	coupled_ode_observer
	Observer for the coupled states. More...
struct	coupled_ode_system
	The coupled_ode_system represents the coupled ode system, which is the base ode and the sensitivities of the base ode (derivatives with respect to the parameters of the base ode). More...
class	coupled_ode_system< F, double, double >
	coupled_ode_system specialization for for known initial values and known parameters. More...
struct	coupled_ode_system< F, double, var >
	The coupled_ode_system template specialization for known initial values and unknown parameters. More...
struct	coupled_ode_system< F, var, double >
	The coupled_ode_system template specialization for unknown initial values and known parameters. More...
struct	coupled_ode_system< F, var, var >
	The coupled_ode_system template specialization for unknown initial values and unknown parameters. More...
class	cov_exp_quad_vari
class	cov_exp_quad_vari< T_x, double, T_l >
class	cvodes_integrator
	Integrator interface for CVODES' ODE solvers (Adams & BDF methods). More...
class	cvodes_ode_data
	CVODES ode data holder object which is used during CVODES integration for CVODES callbacks. More...
struct	digamma_fun
	Structure to wrap digamma() so it can be vectorized. More...
struct	disjunction
	Extends std::false_type when instantiated with zero or more template parameters, all of which extend the std::false_type. More...
struct	disjunction< Cond, Conds... >
class	elewise_division_
	Represents element-wise division in kernel generator expressions. More...
class	elewise_multiplication_
	Represents element-wise multiplication in kernel generator expressions. More...
struct	erf_fun
	Structure to wrap erf() so it can be vectorized. More...
struct	erfc_fun
	Structure to wrap erfc() so that it can be vectorized. More...
struct	exp2_fun
	Structure to wrap exp2() so it can be vectorized. More...
struct	exp_fun
	Structure to wrap exp() so that it can be vectorized. More...
struct	expm1_fun
	Structure to wrap expm1() so that it can be vectorized. More...
struct	fabs_fun
	Structure to wrap fabs() so that it can be vectorized. More...
struct	FixedPointADJac
struct	FixedPointSolver
struct	FixedPointSolver< KinsolFixedPointEnv< F >, fp_jac_type >
struct	floor_fun
	Structure to wrap floor() so that it can be vectorized. More...
struct	fvar
	This template class represents scalars used in forward-mode automatic differentiation, which consist of values and directional derivatives of the specified template type. More...
class	gevv_vvv_vari
class	gp_exp_quad_cov_vari
	This is a subclass of the vari class for precomputed gradients of gp_exp_quad_cov. More...
class	gp_exp_quad_cov_vari< T_x, double, T_l >
	This is a subclass of the vari class for precomputed gradients of gp_exp_quad_cov. More...
class	gp_periodic_cov_vari
	This is a subclass of the vari class for precomputed gradients of gp_periodic_cov. More...
class	gp_periodic_cov_vari< T_x, double, T_l, T_p >
	This is a subclass of the vari class for precomputed gradients of gp_periodic_cov. More...
struct	hybrj_functor_solver
	A functor with the required operators to call Eigen's algebraic solver. More...
class	idas_forward_system
	IDAS DAE system with forward sensitivity calculation. More...
class	idas_integrator
	IDAS DAE integrator. More...
class	idas_system
	IDAS DAE system that contains informtion on residual equation functor, sensitivity residual equation functor, as well as initial conditions. More...
struct	include_summand
	Template metaprogram to calculate whether a summand needs to be included in a proportional (log) probability calculation. More...
struct	include_summand< propto, T >
	true if a term with the specified propto value and subterm types should be included in a proportionality calculation. More...
struct	index_type
	Primary template class for the metaprogram to compute the index type of a container. More...
struct	index_type< T, require_eigen_t< T > >
	Template metaprogram defining typedef for the type of index for an Eigen matrix, vector, or row vector. More...
struct	index_type< T, require_std_vector_t< T > >
	Template metaprogram class to compute the type of index for a standard vector. More...
struct	index_type< T, std::enable_if_t< std::is_pointer< T >::value > >
	Specialization of index_type for pointers.
struct	inv_cloglog_fun
	Structure to wrap inv_cloglog() so that it can be vectorized. More...
struct	inv_fun
	Structure to wrap inv() so that it can be vectorized. More...
struct	inv_logit_fun
	Structure to wrap inv_logit() so that it can be vectorized. More...
struct	inv_Phi_fun
	Structure to wrap inv_Phi() so it can be vectorized. More...
struct	inv_sqrt_fun
	Structure to wrap inv_sqrt() so that it can be vectorized. More...
struct	inv_square_fun
	Structure to wrap inv_square() so that it can be vectorized. More...
struct	is_valid_expression
	Determines whether a type is is a valid kernel generator expression. More...
struct	is_valid_expression_and_not_scalar
	Determines whether a type is non-scalar type that is a valid kernel generator expression. More...
struct	is_valid_expression_and_not_scalar< T, require_matrix_cl_t< T > >
struct	kernel_parts
	Parts of an OpenCL kernel, generated by an expression. More...
struct	kinsol_J_f
	Default Jacobian builder using revser-mode autodiff. More...
class	kinsol_system_data
	KINSOL algebraic system data holder. More...
struct	KinsolFixedPointEnv
	KINSOL algebraic system data holder that handles construction & destruction of SUNDIALS data, as well as auxiliary data that will be used for functor evaluation. More...
class	LDLT_alloc
	This object stores the actual (double typed) LDLT factorization of an Eigen::Matrix<var> along with pointers to its vari's which allow the *ldlt_ functions to save memory. More...
class	LDLT_factor
	LDLT_factor is a thin wrapper on Eigen::LDLT to allow for reusing factorizations and efficient autodiff of things like log determinants and solutions to linear systems. More...
class	LDLT_factor< var, R, C >
	A template specialization of src/stan/math/matrix/LDLT_factor.hpp for var which can be used with all the *_ldlt functions. More...
struct	lgamma_fun
	Structure to wrap lgamma() so that it can be vectorized. More...
class	load_
	Represents an access to a matrix_cl in kernel generator expressions. More...
struct	log10_fun
	Structure to wrap log10() so it can be vectorized. More...
struct	log1m_exp_fun
	Structure to wrap log1m_exp() so it can be vectorized. More...
struct	log1m_fun
	Structure to wrap log1m() so it can be vectorized. More...
struct	log1m_inv_logit_fun
	Structure to wrap log1m_inv_logit() so it can be vectorized. More...
struct	log1p_exp_fun
	Structure to wrap log1m_exp() so that it can be vectorized. More...
struct	log1p_fun
	Structure to wrap log1p() so it can be vectorized. More...
struct	log2_fun
	Structure to wrap log2() so it can be vectorized. More...
struct	log_fun
	Structure to wrap log() so that it can be vectorized. More...
struct	log_inv_logit_fun
	Structure to wrap log_inv_logit() so it can be vectorized. More...
struct	logit_fun
	Structure to wrap logit() so it can be vectorized. More...
class	matrix_cl
class	matrix_cl< T, require_arithmetic_t< T > >
	Represents a matrix on the OpenCL device. More...
class	matrix_cl< T, require_var_t< T > >
class	matrix_exp_action_handler
struct	mpi_cluster
	MPI cluster holds MPI resources and must be initialized only once in any MPI program. More...
struct	mpi_command
	A MPI command object is used to execute code on worker nodes. More...
struct	mpi_distributed_apply
	MPI command template which calls the static method distributed_apply of the given class F. More...
class	mpi_is_in_use
	Exception thrown whenever the MPI resource is busy. More...
class	mpi_parallel_call
	The MPI parallel call class manages the distributed evaluation of a collection of tasks following the map - reduce - combine pattern. More...
class	mpi_stop_listen
	Exception used to stop workers nodes from further listening to commands send from the root. More...
struct	mpi_stop_worker
	MPI command used to stop childs nodes from listening for further commands. More...
class	multiply_mat_vari
	This is a subclass of the vari class for matrix multiplication A * B where A is N by M and B is M by K. More...
class	multiply_mat_vari< double, 1, Ca, Tb, 1 >
	This is a subclass of the vari class for matrix multiplication A * B where A is a double row vector of length M and B is a vector of length M. More...
class	multiply_mat_vari< double, Ra, Ca, Tb, Cb >
	This is a subclass of the vari class for matrix multiplication A * B where A is an N by M matrix of double and B is M by K. More...
class	multiply_mat_vari< Ta, 1, Ca, double, 1 >
	This is a subclass of the vari class for matrix multiplication A * B where A is a row vector of length M and B is a vector of length M of doubles. More...
class	multiply_mat_vari< Ta, 1, Ca, Tb, 1 >
	This is a subclass of the vari class for matrix multiplication A * B where A is 1 by M and B is M by 1. More...
class	multiply_mat_vari< Ta, Ra, Ca, double, Cb >
	This is a subclass of the vari class for matrix multiplication A * B where A is N by M and B is an M by K matrix of doubles. More...
class	name_generator
	Unique name generator for variables used in generated kernels. More...
struct	nlo_functor
	A structure which gets passed to Eigen's dogleg algebraic solver. More...
class	op_ddv_vari
class	op_dv_vari
class	op_dvd_vari
class	op_dvv_vari
class	op_matrix_vari
class	op_v_vari
class	op_vd_vari
class	op_vdd_vari
class	op_vdv_vari
class	op_vector_vari
class	op_vv_vari
class	op_vvd_vari
class	op_vvv_vari
class	opencl_context
	The API to access the methods and values in opencl_context_base. More...
class	opencl_context_base
	The opencl_context_base class represents an OpenCL context in the standard Meyers singleton design pattern. More...
class	operands_and_partials
	This template builds partial derivatives with respect to a set of operands. More...
class	operands_and_partials< Op1, Op2, Op3, Op4, Op5, fvar< Dx > >
	This class builds partial derivatives with respect to a set of operands. More...
class	operands_and_partials< Op1, Op2, Op3, Op4, Op5, var >
	This class builds partial derivatives with respect to a set of operands. More...
class	operation_cl
	Base for all kernel generator operations. More...
class	operation_cl_base
	Non-templated base of operation is needed for easy checking if something is a subclass of operation. More...
class	operation_cl_lhs
	Base for all kernel generator operations that can be used on left hand side of an expression. More...
struct	pass_type
struct	pass_type< double >
struct	pass_type< int >
struct	Phi_approx_fun
	Structure to wrap Phi_approx() so it can be vectorized. More...
struct	Phi_fun
	Structure to wrap Phi() so it can be vectorized. More...
class	precomp_v_vari
class	precomp_vv_vari
class	precomp_vvv_vari
class	precomputed_gradients_vari
	A variable implementation taking a sequence of operands and partial derivatives with respect to the operands. More...
struct	promote_elements
	Struct with static function for elementwise type promotion. More...
struct	promote_elements< Eigen::Matrix< T, R, C >, Eigen::Matrix< S, R, C > >
	Struct with static function for elementwise type promotion.
struct	promote_elements< Eigen::Matrix< T, R, C >, Eigen::Matrix< T, R, C > >
	Struct with static function for elementwise type promotion.
struct	promote_elements< std::vector< T >, std::vector< S > >
	Struct with static function for elementwise type promotion.
struct	promote_elements< std::vector< T >, std::vector< T > >
	Struct with static function for elementwise type promotion.
struct	promote_elements< T, T >
	Struct with static function for elementwise type promotion. More...
struct	promote_scalar_struct
	General struct to hold static function for promoting underlying scalar types. More...
struct	promote_scalar_struct< T, Eigen::Matrix< S, 1,-1 > >
	Struct to hold static function for promoting underlying scalar types.
struct	promote_scalar_struct< T, Eigen::Matrix< S,-1, 1 > >
	Struct to hold static function for promoting underlying scalar types.
struct	promote_scalar_struct< T, Eigen::Matrix< S,-1,-1 > >
	Struct to hold static function for promoting underlying scalar types.
struct	promote_scalar_struct< T, std::vector< S > >
	Struct to hold static function for promoting underlying scalar types.
struct	promote_scalar_struct< T, T >
	Struct to hold static function for promoting underlying scalar types. More...
struct	promote_scalar_type
	Template metaprogram to calculate a type for converting a convertible type. More...
struct	promote_scalar_type< T, Eigen::Matrix< S, 1, Eigen::Dynamic > >
	Template metaprogram to calculate a type for a row vector whose underlying scalar is converted from the second template parameter type to the first.
struct	promote_scalar_type< T, Eigen::Matrix< S, Eigen::Dynamic, 1 > >
	Template metaprogram to calculate a type for a vector whose underlying scalar is converted from the second template parameter type to the first.
struct	promote_scalar_type< T, Eigen::Matrix< S, Eigen::Dynamic, Eigen::Dynamic > >
	Template metaprogram to calculate a type for a matrix whose underlying scalar is converted from the second template parameter type to the first.
struct	promote_scalar_type< T, std::vector< S > >
	Template metaprogram to calculate a type for a container whose underlying scalar is converted from the second template parameter type to the first.
struct	round_fun
	Structure to wrap round() so it can be vectorized. More...
class	scal_squared_distance_dv_vari
class	scal_squared_distance_vd_vari
class	scal_squared_distance_vv_vari
class	scalar_
	Represents a scalar in kernel generator expressions. More...
class	seq_view
class	seq_view< double, std::vector< int > >
class	seq_view< T, Eigen::Matrix< S, 1, Eigen::Dynamic > >
class	seq_view< T, Eigen::Matrix< S, Eigen::Dynamic, 1 > >
class	seq_view< T, Eigen::Matrix< S, Eigen::Dynamic, Eigen::Dynamic > >
class	seq_view< T, std::vector< S > >
class	seq_view< T, std::vector< std::vector< T > > >
class	seq_view< T, std::vector< T > >
struct	sin_fun
	Structure to wrap sin() so it can be vectorized. More...
struct	sinh_fun
	Structure to wrap sinh() so that it can be vectorized. More...
struct	sqrt_fun
	Structure to wrap sqrt() so that it can be vectorized. More...
struct	square_fun
	Structure to wrap square() so that it can be vectorized. More...
class	stack_alloc
	An instance of this class provides a memory pool through which blocks of raw memory may be allocated and then collected simultaneously. More...
struct	store_type
struct	store_type< double >
struct	store_type< int >
class	stored_gradient_vari
	A var implementation that stores the daughter variable implementation pointers and the partial derivative with respect to the result explicitly in arrays constructed on the auto-diff memory stack. More...
class	subtraction_
	Represents subtraction in kernel generator expressions. More...
class	sum_eigen_v_vari
	Class for representing sums with constructors for Eigen. More...
class	sum_v_vari
	Class for sums of variables constructed with standard vectors. More...
struct	system_functor
	A functor that allows us to treat either x or y as the independent variable. More...
struct	tan_fun
	Structure to wrap tan() so that it can be vectorized. More...
struct	tanh_fun
	Structure to wrap tanh() so that it can be vectorized. More...
struct	tgamma_fun
	Structure to wrap tgamma() so that it can be vectorized. More...
struct	trigamma_fun
	Structure to wrap trigamma() so it can be vectorized. More...
struct	trunc_fun
	Structure to wrap trunc() so it can be vectorized. More...
class	unary_function_cl
	Represents a unary function in kernel generator expressions. More...
class	var
	Independent (input) and dependent (output) variables for gradients. More...
class	vari
	The variable implementation base class. More...
class	welford_covar_estimator
class	welford_var_estimator

Typedefs
using	size_type = Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >::Index
	Type for sizes and indexes in an Eigen matrix with double e. More...
using	matrix_fd = Eigen::Matrix< fvar< double >, Eigen::Dynamic, Eigen::Dynamic >
using	matrix_ffd = Eigen::Matrix< fvar< fvar< double > >, Eigen::Dynamic, Eigen::Dynamic >
using	vector_fd = Eigen::Matrix< fvar< double >, Eigen::Dynamic, 1 >
using	vector_ffd = Eigen::Matrix< fvar< fvar< double > >, Eigen::Dynamic, 1 >
using	row_vector_fd = Eigen::Matrix< fvar< double >, 1, Eigen::Dynamic >
using	row_vector_ffd = Eigen::Matrix< fvar< fvar< double > >, 1, Eigen::Dynamic >
using	matrix_fv = Eigen::Matrix< fvar< var >, Eigen::Dynamic, Eigen::Dynamic >
using	matrix_ffv = Eigen::Matrix< fvar< fvar< var >>, Eigen::Dynamic, Eigen::Dynamic >
using	vector_fv = Eigen::Matrix< fvar< var >, Eigen::Dynamic, 1 >
using	vector_ffv = Eigen::Matrix< fvar< fvar< var >>, Eigen::Dynamic, 1 >
using	row_vector_fv = Eigen::Matrix< fvar< var >, 1, Eigen::Dynamic >
using	row_vector_ffv = Eigen::Matrix< fvar< fvar< var >>, 1, Eigen::Dynamic >
template<typename T >
using	as_operation_cl_t = std::conditional_t< std::is_lvalue_reference< T >::value, decltype(as_operation_cl(std::declval< T >())), std::remove_reference_t< decltype(as_operation_cl(std::declval< T >()))>>
	Type that results when converting any valid kernel generator expression into operation. More...
template<typename... Types>
using	require_all_valid_expressions_and_none_scalar_t = require_all_t< is_valid_expression_and_not_scalar< Types >... >
	Enables a template if all given types are non-scalar types that are a valid kernel generator expressions. More...
template<typename... Types>
using	require_all_valid_expressions_t = require_all_t< is_valid_expression< Types >... >
	Enables a template if all given types are are a valid kernel generator expressions. More...
template<typename T >
using	matrix_cl_prim = matrix_cl< T, require_arithmetic_t< T >>
template<typename T >
using	matrix_cl_fp = matrix_cl< T, require_floating_point_t< T >>
using	matrix_d = Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >
	Type for matrix of double values. More...
using	vector_d = Eigen::Matrix< double, Eigen::Dynamic, 1 >
	Type for (column) vector of double values. More...
using	row_vector_d = Eigen::Matrix< double, 1, Eigen::Dynamic >
	Type for (row) vector of double values. More...
template<typename T >
using	index_type_t = typename index_type< T >::type
	Type for sizes and indexes in an Eigen matrix with double e. More...
using	boost_policy_t = boost::math::policies::policy< boost::math::policies::overflow_error< boost::math::policies::errno_on_error >, boost::math::policies::pole_error< boost::math::policies::errno_on_error >, boost::math::policies::promote_double< false > >
	Boost policy that overrides the defaults to match the built-in C++ standard library functions. More...
template<typename T , typename S >
using	promote_scalar_t = typename promote_scalar_type< T, S >::type
using	ChainableStack = AutodiffStackSingleton< vari, chainable_alloc >
using	matrix_v = Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic >
	The type of a matrix holding var values. More...
using	vector_v = Eigen::Matrix< var, Eigen::Dynamic, 1 >
	The type of a (column) vector holding var values. More...
using	row_vector_v = Eigen::Matrix< var, 1, Eigen::Dynamic >
	The type of a row vector holding var values. More...
using	matrix_vi = Eigen::Matrix< vari *, Eigen::Dynamic, Eigen::Dynamic >
	The type of a matrix holding vari* values. More...
using	vector_vi = Eigen::Matrix< vari *, Eigen::Dynamic, 1 >
	The type of a (column) vector holding vari* values. More...
using	row_vector_vi = Eigen::Matrix< vari *, 1, Eigen::Dynamic >
	The type of a row vector holding vari* values. More...

Enumerations
enum	matrix_cl_view { matrix_cl_view::Diagonal = 0, matrix_cl_view::Lower = 1, matrix_cl_view::Upper = 2, matrix_cl_view::Entire = 3 }
enum	TriangularMapCL { TriangularMapCL::UpperToLower = 0, TriangularMapCL::LowerToUpper = 1 }

Functions
template<typename T >
fvar< T >	log_sum_exp (const std::vector< fvar< T > > &v)
template<typename T >
fvar< T >	sum (const std::vector< fvar< T > > &m)
	Return the sum of the entries of the specified standard vector. More...
template<typename T >
std::vector< fvar< T > >	to_fvar (const std::vector< T > &v)
template<typename T >
std::vector< fvar< T > >	to_fvar (const std::vector< T > &v, const std::vector< T > &d)
template<typename T >
const std::vector< fvar< T > > &	to_fvar (const std::vector< fvar< T >> &v)
	Specialization of to_fvar for const fvar input. More...
template<typename T >
std::vector< fvar< T > > &	to_fvar (std::vector< fvar< T >> &v)
	Specialization of to_fvar for non-const fvar input. More...
template<typename T >
fvar< T >	operator+ (const fvar< T > &x1, const fvar< T > &x2)
	Return the sum of the specified forward mode addends. More...
template<typename T >
fvar< T >	operator+ (double x1, const fvar< T > &x2)
	Return the sum of the specified double and forward mode addends. More...
template<typename T >
fvar< T >	operator+ (const fvar< T > &x1, double x2)
	Return the sum of the specified forward mode and double addends. More...
template<typename T >
fvar< T >	operator/ (const fvar< T > &x1, const fvar< T > &x2)
	Return the result of dividing the first argument by the second. More...
template<typename T >
fvar< T >	operator/ (const fvar< T > &x1, double x2)
	Return the result of dividing the first argument by the second. More...
template<typename T >
fvar< T >	operator/ (double x1, const fvar< T > &x2)
	Return the result of dividing the first argument by the second. More...
template<typename T >
bool	operator== (const fvar< T > &x, const fvar< T > &y)
	Return true if the specified variables have equal values as defined by ==. More...
template<typename T >
bool	operator== (const fvar< T > &x, double y)
	Return true if the the first variable has a value equal to the second argument as defined by by ==. More...
template<typename T >
bool	operator== (double x, const fvar< T > &y)
	Return true if the the first argument is equal to the value of the second argument as defined by by ==. More...
template<typename T >
bool	operator> (const fvar< T > &x, const fvar< T > &y)
	Return true if the first argument has a greater value than the second as defined by >. More...
template<typename T >
bool	operator> (const fvar< T > &x, double y)
	Return true if the first argument has a greater value than the second as defined by >. More...
template<typename T >
bool	operator> (double x, const fvar< T > &y)
	Return true if the first argument has a greater value than the second as defined by >. More...
template<typename T >
bool	operator>= (const fvar< T > &x, const fvar< T > &y)
	Return true if the value of the first argument is greater than or equal to that of the second as defined by >=. More...
template<typename T >
bool	operator>= (const fvar< T > &x, double y)
	Return true if the value of the first argument has a value greater than or equal to the second argument as defined by >=. More...
template<typename T >
bool	operator>= (double x, const fvar< T > &y)
	Return true if the first argument is greater than or equal to the value of the second argument as defined by >=. More...
template<typename T >
bool	operator< (const fvar< T > &x, const fvar< T > &y)
	Return true if the first argument has a value less than the value of the second argument as defined by <. More...
template<typename T >
bool	operator< (double x, const fvar< T > &y)
	Return true if the first argument is less than the value of the second argument as defined by <. More...
template<typename T >
bool	operator< (const fvar< T > &x, double y)
	Return true if the first argument has a value less than the second argument as defined by <. More...
template<typename T >
bool	operator<= (const fvar< T > &x, const fvar< T > &y)
	Return true if the first argument has a value less than or equal to the value of the second argument as defined by <=. More...
template<typename T >
bool	operator<= (const fvar< T > &x, double y)
	Return true if the first argument has a value less than or equal to the second argument as defined by <=. More...
template<typename T >
bool	operator<= (double x, const fvar< T > &y)
	Return true if the first argument is less than or equal to the second argument's value as defined by <=. More...
template<typename T >
bool	operator&& (const fvar< T > &x, const fvar< T > &y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename T >
bool	operator&& (const fvar< T > &x, double y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename T >
bool	operator&& (double x, const fvar< T > &y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename T >
bool	operator|| (const fvar< T > &x, const fvar< T > &y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename T >
bool	operator|| (const fvar< T > &x, double y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename T >
bool	operator|| (double x, const fvar< T > &y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename T >
fvar< T >	operator* (const fvar< T > &x, const fvar< T > &y)
	Return the product of the two arguments. More...
template<typename T >
fvar< T >	operator* (double x, const fvar< T > &y)
	Return the product of the two arguments. More...
template<typename T >
fvar< T >	operator* (const fvar< T > &x, double y)
	Return the product of the two arguments. More...
template<typename T >
bool	operator!= (const fvar< T > &x, const fvar< T > &y)
	Return true if the value of the two arguments are not equal as defined by !=. More...
template<typename T >
bool	operator!= (const fvar< T > &x, double y)
	Return true if the value of the first argument is not equal to the second argument as defined by !=. More...
template<typename T >
bool	operator!= (double x, const fvar< T > &y)
	Return true if the first argument is not equal to the value of the second argument as defined by !=. More...
template<typename T >
fvar< T >	operator- (const fvar< T > &x1, const fvar< T > &x2)
	Return the difference of the specified arguments. More...
template<typename T >
fvar< T >	operator- (double x1, const fvar< T > &x2)
	Return the difference of the specified arguments. More...
template<typename T >
fvar< T >	operator- (const fvar< T > &x1, double x2)
	Return the difference of the specified arguments. More...
template<typename T >
fvar< T >	operator- (const fvar< T > &x)
	Return the negation of the specified argument. More...
template<typename T >
bool	operator! (const fvar< T > &x)
	Return the negation of the value of the argument as defined by !. More...
template<typename T >
fvar< T >	operator+ (const fvar< T > &x)
	Returns the argument. More...
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 >	columns_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 >	columns_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 >	columns_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, 1, C >	columns_dot_self (const Eigen::Matrix< fvar< T >, R, C > &x)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, C, C >	crossprod (const Eigen::Matrix< fvar< T >, R, C > &m)
template<typename T , int R, int C>
fvar< T >	determinant (const Eigen::Matrix< fvar< T >, R, C > &m)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	divide (const Eigen::Matrix< fvar< T >, R, C > &v, const fvar< T > &c)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	divide (const Eigen::Matrix< fvar< T >, R, C > &v, double c)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	divide (const Eigen::Matrix< double, R, C > &v, const fvar< T > &c)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	operator/ (const Eigen::Matrix< fvar< T >, R, C > &v, const fvar< T > &c)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	operator/ (const Eigen::Matrix< fvar< T >, R, C > &v, double c)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	operator/ (const Eigen::Matrix< double, R, C > &v, const fvar< T > &c)
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2, size_type &length)
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2, size_type &length)
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2, size_type &length)
template<typename T >
fvar< T >	dot_product (const std::vector< fvar< T > > &v1, const std::vector< fvar< T > > &v2)
template<typename T >
fvar< T >	dot_product (const std::vector< double > &v1, const std::vector< fvar< T > > &v2)
template<typename T >
fvar< T >	dot_product (const std::vector< fvar< T > > &v1, const std::vector< double > &v2)
template<typename T >
fvar< T >	dot_product (const std::vector< fvar< T > > &v1, const std::vector< fvar< T > > &v2, size_type &length)
template<typename T >
fvar< T >	dot_product (const std::vector< double > &v1, const std::vector< fvar< T > > &v2, size_type &length)
template<typename T >
fvar< T >	dot_product (const std::vector< fvar< T > > &v1, const std::vector< double > &v2, size_type &length)
template<typename T >
fvar< T >	dot_product (const fvar< T > *v1, const fvar< T > *v2, size_type length)
	Return dot product of specified pointers up to specified length. More...
template<typename T >
fvar< T >	dot_product (const double *v1, const fvar< T > *v2, size_type length)
	Return dot product of specified pointers up to specified length. More...
template<typename T >
fvar< T >	dot_product (const fvar< T > *v1, const double *v2, size_type length)
	Return dot product of specified pointers up to specified length. More...
template<typename T , int R, int C>
fvar< T >	dot_self (const Eigen::Matrix< fvar< T >, R, C > &v)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	inverse (const Eigen::Matrix< fvar< T >, R, C > &m)
template<typename T , int R, int C>
fvar< T >	log_determinant (const Eigen::Matrix< fvar< T >, R, C > &m)
template<typename T >
Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 >	log_softmax (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &alpha)
template<typename T , int R, int C>
fvar< T >	log_sum_exp (const Eigen::Matrix< fvar< T >, R, C > &v)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_left (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_left (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_left (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
template<int R1, int C1, int R2, int C2, typename T2 >
Eigen::Matrix< fvar< T2 >, R1, C2 >	mdivide_left_ldlt (const LDLT_factor< double, R1, C1 > &A, const Eigen::Matrix< fvar< T2 >, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 >	mdivide_left_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 >	mdivide_left_tri_low (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 >	mdivide_left_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 >	mdivide_right_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right_tri_low (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m, const fvar< T > &c)
template<typename T , int R2, int C2>
Eigen::Matrix< fvar< T >, R2, C2 >	multiply (const Eigen::Matrix< fvar< T >, R2, C2 > &m, double c)
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const Eigen::Matrix< double, R1, C1 > &m, const fvar< T > &c)
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const fvar< T > &c, const Eigen::Matrix< fvar< T >, R1, C1 > &m)
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (double c, const Eigen::Matrix< fvar< T >, R1, C1 > &m)
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const fvar< T > &c, const Eigen::Matrix< double, R1, C1 > &m)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m1, const Eigen::Matrix< fvar< T >, R2, C2 > &m2)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m1, const Eigen::Matrix< double, R2, C2 > &m2)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	multiply (const Eigen::Matrix< double, R1, C1 > &m1, const Eigen::Matrix< fvar< T >, R2, C2 > &m2)
template<typename T , int C1, int R2>
fvar< T >	multiply (const Eigen::Matrix< fvar< T >, 1, C1 > &rv, const Eigen::Matrix< fvar< T >, R2, 1 > &v)
template<typename T , int C1, int R2>
fvar< T >	multiply (const Eigen::Matrix< fvar< T >, 1, C1 > &rv, const Eigen::Matrix< double, R2, 1 > &v)
template<typename T , int C1, int R2>
fvar< T >	multiply (const Eigen::Matrix< double, 1, C1 > &rv, const Eigen::Matrix< fvar< T >, R2, 1 > &v)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, R >	multiply_lower_tri_self_transpose (const Eigen::Matrix< fvar< T >, R, C > &m)
template<typename T >
Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic >	qr_Q (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &m)
template<typename T >
Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic >	qr_R (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &m)
template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix< fvar< T >, CB, CB >	quad_form_sym (const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< double, RB, CB > &B)
template<int RA, int CA, int RB, typename T >
fvar< T >	quad_form_sym (const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< double, RB, 1 > &B)
template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix< fvar< T >, CB, CB >	quad_form_sym (const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, CB > &B)
template<int RA, int CA, int RB, typename T >
fvar< T >	quad_form_sym (const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, 1 > &B)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 >	rows_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 >	rows_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 >	rows_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, 1 >	rows_dot_self (const Eigen::Matrix< fvar< T >, R, C > &x)
template<typename T >
Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 >	softmax (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &alpha)
template<typename T , int R, int C>
fvar< T >	squared_distance (const Eigen::Matrix< fvar< T >, R, C > &v1, const Eigen::Matrix< double, R, C > &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	squared_distance (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<typename T , int R, int C>
fvar< T >	squared_distance (const Eigen::Matrix< double, R, C > &v1, const Eigen::Matrix< fvar< T >, R, C > &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	squared_distance (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<typename T , int R, int C>
fvar< T >	squared_distance (const Eigen::Matrix< fvar< T >, R, C > &v1, const Eigen::Matrix< fvar< T >, R, C > &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	squared_distance (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<typename T , int R, int C>
fvar< T >	sum (const Eigen::Matrix< fvar< T >, R, C > &m)
	Return the sum of the entries of the specified matrix. More...
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, R >	tcrossprod (const Eigen::Matrix< fvar< T >, R, C > &m)
template<int R, int C, typename T >
const Eigen::Matrix< T, R, C > &	to_fvar (const Eigen::Matrix< T, R, C > &m)
	Specialization of to_fvar for const matrices of fvars. More...
template<int R, int C, typename T >
Eigen::Matrix< T, R, C > &	to_fvar (Eigen::Matrix< T, R, C > &m)
	Specialization of to_fvar for non-const matrices of fvars. More...
template<int R, int C>
Eigen::Matrix< fvar< double >, R, C >	to_fvar (const Eigen::Matrix< double, R, C > &m)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	to_fvar (const Eigen::Matrix< T, R, C > &val, const Eigen::Matrix< T, R, C > &deriv)
template<int RA, int CA, int RB, int CB, typename T >
fvar< T >	trace_quad_form (const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, CB > &B)
template<int RA, int CA, int RB, int CB, typename T >
fvar< T >	trace_quad_form (const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< double, RB, CB > &B)
template<int RA, int CA, int RB, int CB, typename T >
fvar< T >	trace_quad_form (const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, CB > &B)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	unit_vector_constrain (const Eigen::Matrix< fvar< T >, R, C > &y)
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	unit_vector_constrain (const Eigen::Matrix< fvar< T >, R, C > &y, fvar< T > &lp)
template<typename T , typename F >
void	gradient (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &grad_fx)
	Calculate the value and the gradient of the specified function at the specified argument. More...
template<typename T , typename F >
void	hessian (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &grad, Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &H)
	Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in time O(N^3) time and O(N^2) space. More...
template<typename T , typename F >
void	jacobian (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, Eigen::Matrix< T, Eigen::Dynamic, 1 > &fx, Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &J)
template<typename T >
fvar< T >	abs (const fvar< T > &x)
template<typename T >
fvar< T >	acos (const fvar< T > &x)
template<typename T >
fvar< T >	acosh (const fvar< T > &x)
template<typename T >
fvar< T >	asin (const fvar< T > &x)
template<typename T >
fvar< T >	asinh (const fvar< T > &x)
template<typename T >
fvar< T >	atan (const fvar< T > &x)
template<typename T >
fvar< T >	atan2 (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	atan2 (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	atan2 (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	atanh (const fvar< T > &x)
	Return inverse hyperbolic tangent of specified value. More...
template<typename T >
fvar< T >	bessel_first_kind (int v, const fvar< T > &z)
template<typename T >
fvar< T >	bessel_second_kind (int v, const fvar< T > &z)
template<typename T >
fvar< T >	beta (const fvar< T > &x1, const fvar< T > &x2)
	Return fvar with the beta function applied to the specified arguments and its gradient. More...
template<typename T >
fvar< T >	beta (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	beta (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	binary_log_loss (int y, const fvar< T > &y_hat)
template<typename T >
fvar< T >	cbrt (const fvar< T > &x)
	Return cube root of specified argument. More...
template<typename T >
fvar< T >	ceil (const fvar< T > &x)
template<typename T >
fvar< T >	cos (const fvar< T > &x)
template<typename T >
fvar< T >	cosh (const fvar< T > &x)
template<typename T >
fvar< T >	digamma (const fvar< T > &x)
	Return the derivative of the log gamma function at the specified argument. More...
template<typename T >
fvar< T >	erf (const fvar< T > &x)
template<typename T >
fvar< T >	erfc (const fvar< T > &x)
template<typename T >
fvar< T >	exp (const fvar< T > &x)
template<typename T >
fvar< T >	exp2 (const fvar< T > &x)
template<typename T >
fvar< T >	expm1 (const fvar< T > &x)
template<typename T >
fvar< T >	fabs (const fvar< T > &x)
template<typename T >
fvar< T >	falling_factorial (const fvar< T > &x, int n)
	Return autodiff variable with the gradient and result of the falling factorial function applied to the inputs. More...
template<typename T >
fvar< T >	fdim (const fvar< T > &x, const fvar< T > &y)
	Return the positive difference of the specified values (C++11). More...
template<typename T >
fvar< T >	fdim (const fvar< T > &x, double y)
	Return the positive difference of the specified values (C++11). More...
template<typename T >
fvar< T >	fdim (double x, const fvar< T > &y)
	Return the positive difference of the specified values (C++11). More...
template<typename T >
fvar< T >	floor (const fvar< T > &x)
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type >	fma (const fvar< T1 > &x1, const fvar< T2 > &x2, const fvar< T3 > &x3)
	The fused multiply-add operation (C99). More...
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type >	fma (const T1 &x1, const fvar< T2 > &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type >	fma (const fvar< T1 > &x1, const T2 &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type >	fma (const fvar< T1 > &x1, const fvar< T2 > &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type >	fma (const T1 &x1, const T2 &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type >	fma (const fvar< T1 > &x1, const T2 &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type >	fma (const T1 &x1, const fvar< T2 > &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...
template<typename T >
fvar< T >	fmax (const fvar< T > &x1, const fvar< T > &x2)
	Return the greater of the two specified arguments. More...
template<typename T >
fvar< T >	fmax (double x1, const fvar< T > &x2)
	Return the greater of the two specified arguments. More...
template<typename T >
fvar< T >	fmax (const fvar< T > &x1, double x2)
	Return the greater of the two specified arguments. More...
template<typename T >
fvar< T >	fmin (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	fmin (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	fmin (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	fmod (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	fmod (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	fmod (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	gamma_p (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	gamma_p (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	gamma_p (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	gamma_q (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	gamma_q (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	gamma_q (double x1, const fvar< T > &x2)
template<typename T >
void	grad_inc_beta (fvar< T > &g1, fvar< T > &g2, fvar< T > a, fvar< T > b, fvar< T > z)
	Gradient of the incomplete beta function beta(a, b, z) with respect to the first two arguments. More...
template<typename T >
fvar< T >	hypot (const fvar< T > &x1, const fvar< T > &x2)
	Return the length of the hypoteneuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11). More...
template<typename T >
fvar< T >	hypot (const fvar< T > &x1, double x2)
	Return the length of the hypoteneuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11). More...
template<typename T >
fvar< T >	hypot (double x1, const fvar< T > &x2)
	Return the length of the hypoteneuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11). More...
template<typename T >
fvar< T >	inc_beta (const fvar< T > &a, const fvar< T > &b, const fvar< T > &x)
template<typename T >
fvar< T >	inc_beta (double a, const fvar< T > &b, const fvar< T > &x)
template<typename T >
fvar< T >	inc_beta (const fvar< T > &a, double b, const fvar< T > &x)
template<typename T >
fvar< T >	inc_beta (const fvar< T > &a, const fvar< T > &b, double x)
template<typename T >
fvar< T >	inc_beta (double a, double b, const fvar< T > &x)
template<typename T >
fvar< T >	inc_beta (const fvar< T > &a, double b, double x)
template<typename T >
fvar< T >	inc_beta (double a, const fvar< T > &b, double x)
template<typename T >
fvar< T >	inv (const fvar< T > &x)
template<typename T >
fvar< T >	inv_cloglog (const fvar< T > &x)
template<typename T >
fvar< T >	inv_logit (const fvar< T > &x)
	Returns the inverse logit function applied to the argument. More...
template<typename T >
fvar< T >	inv_Phi (const fvar< T > &p)
template<typename T >
fvar< T >	inv_sqrt (const fvar< T > &x)
template<typename T >
fvar< T >	inv_square (const fvar< T > &x)
template<typename T >
int	is_inf (const fvar< T > &x)
	Returns 1 if the input's value is infinite and 0 otherwise. More...
template<typename T , require_fvar_t< T > ...>
bool	is_nan (T &&x)
	Returns 1 if the input's value is NaN and 0 otherwise. More...
template<typename T >
fvar< T >	lbeta (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	lbeta (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	lbeta (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	ldexp (const fvar< T > &a, int b)
	Returns the product of a (the significand) times 2 to power b (the exponent). More...
template<typename T >
fvar< T >	lgamma (const fvar< T > &x)
	Return the natural logarithm of the gamma function applied to the specified argument. More...
template<typename T >
fvar< typename stan::return_type< T, int >::type >	lmgamma (int x1, const fvar< T > &x2)
template<typename T >
fvar< T >	log (const fvar< T > &x)
template<typename T >
fvar< T >	log10 (const fvar< T > &x)
template<typename T >
fvar< T >	log1m (const fvar< T > &x)
template<typename T >
fvar< T >	log1m_exp (const fvar< T > &x)
	Return the natural logarithm of one minus the exponentiation of the specified argument. More...
template<typename T >
fvar< T >	log1m_inv_logit (const fvar< T > &x)
	Return the natural logarithm of one minus the inverse logit of the specified argument. More...
template<typename T >
fvar< T >	log1p (const fvar< T > &x)
template<typename T >
fvar< T >	log1p_exp (const fvar< T > &x)
template<typename T >
fvar< T >	log2 (const fvar< T > &x)
	Return the base two logarithm of the specified argument. More...
template<typename T >
fvar< T >	log_diff_exp (const fvar< T > &x1, const fvar< T > &x2)
template<typename T1 , typename T2 >
fvar< T2 >	log_diff_exp (const T1 &x1, const fvar< T2 > &x2)
template<typename T1 , typename T2 >
fvar< T1 >	log_diff_exp (const fvar< T1 > &x1, const T2 &x2)
template<typename T >
fvar< T >	log_falling_factorial (const fvar< T > &x, const fvar< T > &n)
template<typename T >
fvar< T >	log_falling_factorial (double x, const fvar< T > &n)
template<typename T >
fvar< T >	log_falling_factorial (const fvar< T > &x, double n)
template<typename T >
fvar< T >	log_inv_logit (const fvar< T > &x)
template<typename T >
fvar< T >	log_inv_logit_diff (const fvar< T > &x, const fvar< T > &y)
	Returns fvar with the natural logarithm of the difference of the inverse logits of the specified arguments and its gradients. More...
template<typename T >
fvar< T >	log_inv_logit_diff (const fvar< T > &x, double y)
template<typename T >
fvar< T >	log_inv_logit_diff (double x, const fvar< T > &y)
template<typename T_theta , typename T_lambda1 , typename T_lambda2 , int N>
void	log_mix_partial_helper (const T_theta &theta, const T_lambda1 &lambda1, const T_lambda2 &lambda2, promote_args_t< T_theta, T_lambda1, T_lambda2 >(&partials_array)[N])
template<typename T >
fvar< T >	log_mix (const fvar< T > &theta, const fvar< T > &lambda1, const fvar< T > &lambda2)
	Return the log mixture density with specified mixing proportion and log densities and its derivative at each. More...
template<typename T , typename P , require_all_arithmetic_t< P > ...>
fvar< T >	log_mix (const fvar< T > &theta, const fvar< T > &lambda1, P lambda2)
template<typename T , typename P , require_all_arithmetic_t< P > ...>
fvar< T >	log_mix (const fvar< T > &theta, P lambda1, const fvar< T > &lambda2)
template<typename T , typename P , require_all_arithmetic_t< P > ...>
fvar< T >	log_mix (P theta, const fvar< T > &lambda1, const fvar< T > &lambda2)
template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > ...>
fvar< T >	log_mix (const fvar< T > &theta, P1 lambda1, P2 lambda2)
template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > ...>
fvar< T >	log_mix (P1 theta, const fvar< T > &lambda1, P2 lambda2)
template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > ...>
fvar< T >	log_mix (P1 theta, P2 lambda1, const fvar< T > &lambda2)
template<typename T >
fvar< T >	log_rising_factorial (const fvar< T > &x, const fvar< T > &n)
template<typename T >
fvar< T >	log_rising_factorial (const fvar< T > &x, double n)
template<typename T >
fvar< T >	log_rising_factorial (double x, const fvar< T > &n)
template<typename T >
fvar< T >	log_sum_exp (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	log_sum_exp (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	log_sum_exp (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	logit (const fvar< T > &x)
template<typename T >
fvar< T >	modified_bessel_first_kind (int v, const fvar< T > &z)
template<typename T >
fvar< T >	modified_bessel_second_kind (int v, const fvar< T > &z)
template<typename T >
fvar< T >	multiply_log (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	multiply_log (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	multiply_log (const fvar< T > &x1, double x2)
template<typename T >
fvar< T >	owens_t (const fvar< T > &x1, const fvar< T > &x2)
	Return Owen's T function applied to the specified arguments. More...
template<typename T >
fvar< T >	owens_t (double x1, const fvar< T > &x2)
	Return Owen's T function applied to the specified arguments. More...
template<typename T >
fvar< T >	owens_t (const fvar< T > &x1, double x2)
	Return Owen's T function applied to the specified arguments. More...
template<typename T >
fvar< T >	Phi (const fvar< T > &x)
template<typename T >
fvar< T >	Phi_approx (const fvar< T > &x)
	Return an approximation of the unit normal cumulative distribution function (CDF). More...
template<typename T >
fvar< T >	pow (const fvar< T > &x1, const fvar< T > &x2)
template<typename T >
fvar< T >	pow (double x1, const fvar< T > &x2)
template<typename T >
fvar< T >	pow (const fvar< T > &x1, double x2)
template<typename T >
double	primitive_value (const fvar< T > &v)
	Return the primitive value of the specified forward-mode autodiff variable. More...
template<typename T >
fvar< T >	rising_factorial (const fvar< T > &x, int n)
	Return autodiff variable with the gradient and result of the rising factorial function applied to the inputs. More...
template<typename T >
fvar< T >	round (const fvar< T > &x)
	Return the closest integer to the specified argument, with halfway cases rounded away from zero. More...
template<typename T >
fvar< T >	sin (const fvar< T > &x)
template<typename T >
fvar< T >	sinh (const fvar< T > &x)
template<typename T >
fvar< T >	sqrt (const fvar< T > &x)
template<typename T >
fvar< T >	square (const fvar< T > &x)
template<typename T >
fvar< T >	tan (const fvar< T > &x)
template<typename T >
fvar< T >	tanh (const fvar< T > &x)
template<typename T >
fvar< T >	tgamma (const fvar< T > &x)
	Return the result of applying the gamma function to the specified argument. More...
template<typename T >
fvar< T >	to_fvar (const T &x)
template<typename T >
const fvar< T > &	to_fvar (const fvar< T > &x)
	Specialization of to_fvar for const fvars. More...
template<typename T >
fvar< T > &	to_fvar (fvar< T > &x)
	Specialization of to_fvar for non-const fvars. More...
template<typename T >
fvar< T >	trigamma (const fvar< T > &u)
	Return the value of the trigamma function at the specified argument (i.e., the second derivative of the log Gamma function at the specified argument). More...
template<typename T >
fvar< T >	trunc (const fvar< T > &x)
	Return the nearest integral value that is not larger in magnitude than the specified argument. More...
template<typename T >
T	value_of (const fvar< T > &v)
	Return the value of the specified variable. More...
template<typename T >
double	value_of_rec (const fvar< T > &v)
	Return the value of the specified variable. More...
template<typename T >
bool	is_aligned (T *ptr, unsigned int bytes_aligned)
	Return true if the specified pointer is aligned on the number of bytes. More...
template<typename T , typename F >
void	derivative (const F &f, const T &x, T &fx, T &dfx_dx)
	Return the derivative of the specified univariate function at the specified argument. More...
template<typename F >
void	finite_diff_grad_hessian (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::MatrixXd &hess, std::vector< Eigen::MatrixXd > &grad_hess_fx, double epsilon=1e-04)
	Calculate the value and the gradient of the hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference. More...
template<typename F >
void	finite_diff_grad_hessian_auto (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::MatrixXd &hess, std::vector< Eigen::MatrixXd > &grad_hess_fx)
	Calculate the value, Hessian, and the gradient of the Hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference. More...
template<typename F >
void	grad_hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &H, std::vector< Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > > &grad_H)
	Calculate the value, the Hessian, and the gradient of the Hessian of the specified function at the specified argument. More...
template<typename F >
void	grad_tr_mat_times_hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &M, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad_tr_MH)
template<typename T1 , typename T2 , typename F >
void	gradient_dot_vector (const F &f, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &v, T1 &fx, T1 &grad_fx_dot_v)
template<typename F >
void	hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &H)
	Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in O(N^2) time and O(N^2) space. More...
template<typename F >
void	hessian_times_vector (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &v, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &Hv)
template<typename T , typename F >
void	hessian_times_vector (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &Hv)
template<typename T , typename F >
void	partial_derivative (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, int n, T &fx, T &dfx_dxn)
	Return the partial derivative of the specified multiivariate function at the specified argument. More...
template<typename Mat , typename Mat_scalar = scalar_type_t<Mat>, require_eigen_vt< std::is_arithmetic, Mat > ...>
matrix_cl< Mat_scalar >	to_matrix_cl (Mat &&src)
	Copies the source Eigen matrix to the destination matrix that is stored on the OpenCL device. More...
template<typename Vec , typename Vec_scalar = scalar_type_t<Vec>, require_std_vector_vt< std::is_arithmetic, Vec > ...>
matrix_cl< Vec_scalar >	to_matrix_cl (Vec &&src)
	Copies the source std::vector to the destination matrix that is stored on the OpenCL device. More...
template<int R = Eigen::Dynamic, int C = Eigen::Dynamic, typename T , typename = require_arithmetic_t<T>>
Eigen::Matrix< T, R, C >	from_matrix_cl (const matrix_cl< T > &src)
	Copies the source matrix that is stored on the OpenCL device to the destination Eigen matrix. More...
template<typename T , typename = require_arithmetic_t<T>>
std::vector< T >	packed_copy (const matrix_cl< T > &src)
	Packs the flat triangular matrix on the OpenCL device and copies it to the std::vector. More...
template<matrix_cl_view matrix_view, typename Vec , typename Vec_scalar = scalar_type_t<Vec>, require_vector_vt< std::is_arithmetic, Vec > ...>
matrix_cl< Vec_scalar >	packed_copy (Vec &&src, int rows)
	Copies the packed triangular matrix from the source std::vector to an OpenCL buffer and unpacks it to a flat matrix on the OpenCL device. More...
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	copy_cl (const matrix_cl< T > &src)
	Copies the source matrix to the destination matrix. More...
template<typename T , typename = require_arithmetic_t<T>>
T	from_matrix_cl_error_code (const matrix_cl< T > &src)
	Copy A 1 by 1 source matrix from the Device to the host. More...
template<typename T , typename = require_arithmetic_t<std::decay_t<T>>>
matrix_cl< std::decay_t< T > >	to_matrix_cl (T &&src)
	Copy an arithmetic type to the device. More...
template<matrix_cl_view matrix_view = matrix_cl_view::Entire, typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	copy_triangular (const matrix_cl< T > &src)
	Copies the lower or upper triangular of the source matrix to the destination matrix. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
matrix_cl< return_type_t< T1, T2 > >	diagonal_multiply (const matrix_cl< T1 > &A, const T2 scalar)
	Multiplies the diagonal of a matrix on the OpenCL device with the specified scalar. More...
template<typename T , typename = require_arithmetic_t<T>>
void	check_diagonal_zeros (const char *function, const char *name, const matrix_cl< T > &y)
	Check if the matrix_cl has zeros on the diagonal. More...
template<typename T >
void	check_invalid_matrix_view (const char *function, const char *name, const matrix_cl< T > &A, const matrix_cl_view invalid_view)
	Check if the matrix_cl has an invalid view. More...
template<typename T >
void	check_mat_size_one (const char *function, const char *name, const matrix_cl< T > &A)
	Check if the matrix_cl has a single element. More...
template<typename T >
void	check_mat_not_size_one (const char *function, const char *name, const matrix_cl< T > &A)
	Check if the matrix_cl has a single element. More...
template<typename T >
void	check_matching_dims (const char *function, const char *name1, const matrix_cl< T > &y1, const char *name2, const matrix_cl< T > &y2)
	Check if two matrix_cls have the same dimensions. More...
template<typename T , typename = require_floating_point_t<T>>
void	check_nan (const char *function, const char *name, const matrix_cl< T > &y)
	Check if the matrix_cl has NaN values. More...
void	check_opencl_error (const char *function, const cl::Error &e)
	Throws the domain error with specifying the OpenCL error that occured. More...
template<typename T >
void	check_square (const char *function, const char *name, const matrix_cl< T > &y)
	Check if the matrix_cl is square. More...
template<typename T , typename = require_arithmetic_t<T>>
void	check_symmetric (const char *function, const char *name, const matrix_cl< T > &y)
	Check if the matrix_cl is symmetric. More...
template<typename T >
void	check_triangular (const char *function, const char *name, const matrix_cl< T > &A)
	Check if the matrix_cl is either upper triangular or lower triangular. More...
template<typename T >
void	check_vector (const char *function, const char *name, const matrix_cl< T > &x)
	Check if the matrix is either a row vector or column vector. More...
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	identity (int rows_cols)
	Returns the identity matrix stored on the OpenCL device. More...
template<typename T_operation , typename = std::enable_if_t<std::is_base_of< operation_cl_base, std::remove_reference_t<T_operation>>::value>>
T_operation &&	as_operation_cl (T_operation &&a)
	Converts any valid kernel generator expression into an operation. More...
template<typename T_scalar , typename = require_arithmetic_t<T_scalar>>
scalar_< T_scalar >	as_operation_cl (const T_scalar a)
	Converts any valid kernel generator expression into an operation. More...
template<typename T_matrix_cl , typename = require_matrix_cl_t<T_matrix_cl>>
load_< T_matrix_cl >	as_operation_cl (T_matrix_cl &&a)
	Converts any valid kernel generator expression into an operation. More...
template<typename T_a , typename T_b , typename = require_all_valid_expressions_t<T_a, T_b>>
addition_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator+ (T_a &&a, T_b &&b)
	Addition of two kernel generator expressions. More...
template<typename T_a , typename T_b , typename = require_all_valid_expressions_t<T_a, T_b>>
subtraction_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	operator- (T_a &&a, T_b &&b)
	Subtraction of two kernel generator expressions. More...
template<typename T_a , typename T_b >
elewise_multiplication_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	elewise_multiplication (T_a &&a, T_b &&b)
	Element-wise multiplication of two kernel generator expressions. More...
template<typename T_a , typename T_b , typename = require_arithmetic_t<T_a>, typename = require_all_valid_expressions_t<T_b>>
elewise_multiplication_< scalar_< T_a >, as_operation_cl_t< T_b > >	operator* (T_a &&a, T_b &&b)
	Multiplication of a scalar and a kernel generator expression. More...
template<typename T_a , typename T_b , typename = require_all_valid_expressions_t<T_a>, typename = require_arithmetic_t<T_b>>
elewise_multiplication_< as_operation_cl_t< T_a >, scalar_< T_b > >	operator* (T_a &&a, const T_b b)
	Multiplication of a kernel generator expression and a scalar. More...
template<typename T_a , typename T_b , typename = require_all_valid_expressions_and_none_scalar_t<T_a, T_b>>
matrix_cl< double >	operator* (const T_a &a, const T_b &b)
	Matrix multiplication of two kernel generator expressions. More...
template<typename T_a , typename T_b , typename = require_all_valid_expressions_t<T_a, T_b>>
elewise_division_< as_operation_cl_t< T_a >, as_operation_cl_t< T_b > >	elewise_division (T_a &&a, T_b &&b)
	Element-wise division of two kernel generator expressions. More...
const matrix_cl_view	either (const matrix_cl_view left_view, const matrix_cl_view right_view)
	Determines which parts are nonzero in any of the input views. More...
const matrix_cl_view	both (const matrix_cl_view left_view, const matrix_cl_view right_view)
	Determines which parts are nonzero in both input views. More...
bool	contains_nonzero (const matrix_cl_view view, const matrix_cl_view part)
	Check whether a view contains certain nonzero part. More...
const matrix_cl_view	transpose (const matrix_cl_view view)
	Transposes a view - swaps lower and upper parts. More...
const matrix_cl_view	invert (const matrix_cl_view view)
	Inverts a view. More...
matrix_cl_view	from_eigen_uplo_type (Eigen::UpLoType eigen_type)
	Creates a view from Eigen::UpLoType. More...
int	either (int left_view, int right_view)
	Determines which parts are nonzero in any of the input views. More...
int	both (int left_view, int right_view)
	Determines which parts are nonzero in both input views. More...
bool	contains_nonzero (int view, int part)
	Check whether a view contains certain nonzero part. More...
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	multiply_transpose (const matrix_cl< T > &A)
	Computes the product of a square OpenCL matrix with its transpose. More...
template<bool propto, typename T_alpha , typename T_beta >
return_type_t< T_alpha, T_beta >	bernoulli_logit_glm_lpmf (const matrix_cl< int > &y_cl, const matrix_cl< double > &x_cl, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Bernoulli distribution and logit link function. More...
template<typename T_alpha , typename T_beta >
return_type_t< T_beta, T_alpha >	bernoulli_logit_glm_lpmf (const matrix_cl< int > &y, const matrix_cl< double > &x, const T_alpha &alpha, const T_beta &beta)
template<bool propto, typename T_alpha_scalar , typename T_beta_scalar >
return_type_t< T_alpha_scalar, T_beta_scalar >	categorical_logit_glm_lpmf (const matrix_cl< int > &y_cl, const matrix_cl< double > &x_cl, const Eigen::Matrix< T_alpha_scalar, Eigen::Dynamic, 1 > &alpha, const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, Eigen::Dynamic > &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with categorical distribution and logit (softmax) link function. More...
template<typename T_alpha_scalar , typename T_beta_scalar >
return_type_t< T_alpha_scalar, T_beta_scalar >	categorical_logit_glm_lpmf (const matrix_cl< int > &y, const matrix_cl< double > &x, const Eigen::Matrix< T_alpha_scalar, Eigen::Dynamic, 1 > &alpha, const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, Eigen::Dynamic > &beta)
template<typename T , typename = require_floating_point_t<T>>
matrix_cl< T >	cholesky_decompose (matrix_cl< T > &A)
	Returns the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix on the OpenCL device. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
void	divide_columns (const matrix_cl< T1 > &A, const matrix_cl< T2 > &B)
	Divides each column of a matrix by a vector. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
void	divide_columns (const matrix_cl< T1 > &A, const T2 &divisor)
	Divides each column of a matrix by a scalar. More...
template<typename T1 , typename T2 , typename T3 , typename = require_all_arithmetic_t<T1, T2, T3>>
matrix_cl< return_type_t< T1, T2, T3 > >	gp_exp_quad_cov (const matrix_cl< T1 > &x, const T2 sigma, const T3 length_scale)
	Squared exponential kernel on the GPU. More...
template<typename T1 , typename T2 , typename T3 , typename T4 , typename = require_all_arithmetic_t<T1, T2, T3, T4>>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	gp_exp_quad_cov (const matrix_cl< T1 > &x, const matrix_cl< T2 > &y, const T3 sigma, const T4 length_scale)
	Squared exponential kernel on the GPU. More...
template<typename T1 , typename T2 , typename = require_all_floating_point_t<T1, T2>>
matrix_cl< return_type_t< T1, T2 > >	mdivide_left_tri_low (const matrix_cl< T1 > &A, const matrix_cl< T2 > &b)
	Returns the solution of the system Ax=b when A is lower triangular. More...
template<typename T , typename = require_all_floating_point_t<T>>
matrix_cl< T >	mdivide_left_tri_low (const matrix_cl< T > &A)
	Returns the solution of the system Ax=b when A is triangular and b=I. More...
template<typename T1 , typename T2 , typename = require_all_floating_point_t<T1, T2>>
matrix_cl< return_type_t< T1, T2 > >	mdivide_right_tri_low (const matrix_cl< T2 > &b, const matrix_cl< T1 > &A)
	Returns the solution of the system Ax=b where A is a lower triangular matrix. More...
template<bool propto, typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_lpmf (const matrix_cl< int > &y_cl, const matrix_cl< double > &x_cl, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
	Returns the log PMF of the Generalized Linear Model (GLM) with Negative-Binomial-2 distribution and log link function. More...
template<typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_lpmf (const matrix_cl< int > &y, const matrix_cl< double > &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
template<bool propto, typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_alpha, T_beta, T_scale >	normal_id_glm_lpdf (const matrix_cl< double > &y_cl, const matrix_cl< double > &x_cl, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
	Returns the log PDF of the Generalized Linear Model (GLM) with Normal distribution and id link function. More...
template<typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_alpha, T_beta, T_scale >	normal_id_glm_lpdf (const matrix_cl< double > &y, const matrix_cl< double > &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
template<bool propto, typename T_beta_scalar , typename T_cuts_scalar >
stan::return_type_t< T_beta_scalar, T_cuts_scalar >	ordered_logistic_glm_lpmf (const matrix_cl< int > &y_cl, const matrix_cl< double > &x_cl, const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, 1 > &beta, const Eigen::Matrix< T_cuts_scalar, Eigen::Dynamic, 1 > &cuts)
	Returns the log PMF of the ordinal regression Generalized Linear Model (GLM). More...
template<typename T_beta_scalar , typename T_cuts_scalar >
return_type< T_beta_scalar, T_cuts_scalar >::type	ordered_logistic_glm_lpmf (const matrix_cl< int > &y, const matrix_cl< double > &x, const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, 1 > &beta, const Eigen::Matrix< T_cuts_scalar, Eigen::Dynamic, 1 > &cuts)
template<bool propto, typename T_alpha , typename T_beta >
return_type_t< T_alpha, T_beta >	poisson_log_glm_lpmf (const matrix_cl< int > &y_cl, const matrix_cl< double > &x_cl, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Poisson distribution and log link function. More...
template<typename T_alpha , typename T_beta >
return_type_t< T_alpha, T_beta >	poisson_log_glm_lpmf (const matrix_cl< int > &y, const matrix_cl< double > &x, const T_alpha &alpha, const T_beta &beta)
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	rep_matrix (const matrix_cl< T > &x, int n, int m)
	Creates a matrix_cl by replicating the value of the only element in the input 1x1 matrix_cl. More...
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	rep_matrix (const matrix_cl< T > &x, int m)
	Creates a matrix_cl by replicating the input vector or row_vector. More...
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	rep_row_vector (const matrix_cl< T > &x, int m)
	Creates a matrix_cl representing a row_vector by replicating the value of the only element in the input 1x1 matrix_cl. More...
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	rep_vector (const matrix_cl< T > &x, int m)
	Creates a matrix_cl representing a vector by replicating the value of the only element in the input 1x1 matrix_cl. More...
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	transpose (const matrix_cl< T > &src)
	Takes the transpose of the matrix on the OpenCL device. More...
template<typename T , int R, int C, require_var_t< T > ...>
matrix_cl< T >	to_matrix_cl (const Eigen::Matrix< T, R, C > &src)
	Copies the source Eigen matrix of vars to the destination matrix that is stored on the OpenCL device. More...
template<int R = Eigen::Dynamic, int C = Eigen::Dynamic, typename T , require_var_t< T > ...>
Eigen::Matrix< double, R, C >	from_matrix_cl (const matrix_cl< T > &src)
	Copies the adjoint of the source matrix of vars that is stored on the OpenCL device to the destination Eigen matrix. More...
template<typename T , typename = require_var_t<T>>
std::vector< double >	packed_copy (const matrix_cl< T > &src)
	Packs the flat triangular matrix on the OpenCL device and copies the adjoint values to the std::vector. More...
template<matrix_cl_view matrix_view>
matrix_cl< var >	packed_copy (vari **src, int rows)
	Copies the packed triangular matrix from the source std::vector to an OpenCL buffer and unpacks it to a flat matrix on the OpenCL device. More...
template<matrix_cl_view matrix_view = matrix_cl_view::Entire, typename T , typename = require_floating_point_t<T>>
matrix_cl< T >	tri_inverse (const matrix_cl< T > &A)
	Computes the inverse of a triangular matrix. More...
template<typename T_y1 , typename T_y2 >
void	check_matching_sizes (const char *function, const char *name1, const T_y1 &y1, const char *name2, const T_y2 &y2)
	Check if two structures at the same size. More...
template<typename T_y >
void	check_nonzero_size (const char *function, const char *name, const T_y &y)
	Check if the specified matrix/vector is of non-zero size. More...
template<typename T_y >
void	check_ordered (const char *function, const char *name, const std::vector< T_y > &y)
	Check if the specified vector is sorted into strictly increasing order. More...
template<typename T_y1 , typename T_y2 >
bool	is_matching_size (const T_y1 &y1, const T_y2 &y2)
	Return true if two structures are the same size. More...
template<typename T_y >
bool	is_nonzero_size (const T_y &y)
	Returns true if the specified matrix/vector is size nonzero. More...
template<typename T_y >
bool	is_ordered (const std::vector< T_y > &y)
	Return true if the vector is sorted into strictly increasing order. More...
double	dot (const std::vector< double > &x, const std::vector< double > &y)
double	dot_self (const std::vector< double > &x)
template<typename T , typename S >
void	fill (std::vector< T > &x, const S &y)
	Fill the specified container with the specified value. More...
template<typename Vector >
void	inverse_softmax (const Vector &simplex, Vector &y)
	Writes the inverse softmax of the simplex argument into the second argument. More...
double	log_sum_exp (const std::vector< double > &x)
	Return the log of the sum of the exponentiated values of the specified sequence of values. More...
template<typename T >
std::vector< T >	rep_array (const T &x, int n)
template<typename T >
std::vector< std::vector< T > >	rep_array (const T &x, int m, int n)
template<typename T >
std::vector< std::vector< std::vector< T > > >	rep_array (const T &x, int k, int m, int n)
void	scaled_add (std::vector< double > &x, const std::vector< double > &y, double lambda)
template<typename T >
std::vector< T >	sort_asc (std::vector< T > xs)
	Return the specified standard vector in ascending order. More...
template<typename T >
std::vector< T >	sort_desc (std::vector< T > xs)
	Return the specified standard vector in descending order. More...
template<typename T >
T	sum (const std::vector< T > &xs)
	Return the sum of the values in the specified standard vector. More...
template<typename T >
std::vector< typename child_type< T >::type >	value_of (const std::vector< T > &x)
	Convert a std::vector of type T to a std::vector of child_type<T>::type. More...
const std::vector< double > &	value_of (const std::vector< double > &x)
	Return the specified argument. More...
const std::vector< int > &	value_of (const std::vector< int > &x)
	Return the specified argument. More...
template<typename T >
std::vector< double >	value_of_rec (const std::vector< T > &x)
	Convert a std::vector of type T to a std::vector of doubles. More...
const std::vector< double > &	value_of_rec (const std::vector< double > &x)
	Return the specified argument. More...
template<typename T >
const std::vector< T > &	vec_concat (const std::vector< T > &v1)
	Ends the recursion to extract the event stack. More...
template<typename T , typename... Args>
const std::vector< T >	vec_concat (const std::vector< T > &v1, const Args...args)
	Gets the event stack from a vector of events and other arguments. More...
template<typename F >
double	integrate (const F &f, double a, double b, double relative_tolerance)
	Integrate a single variable function f from a to b to within a specified relative tolerance. More...
template<typename F >
double	integrate_1d (const F &f, const double a, const double b, const std::vector< double > &theta, const std::vector< double > &x_r, const std::vector< int > &x_i, std::ostream *msgs, const double relative_tolerance=std::sqrt(EPSILON))
	Compute the integral of the single variable function f from a to b to within a specified relative tolerance. More...
template<typename F , typename T1 , typename T2 , typename T_t0 , typename T_ts >
std::vector< std::vector< return_type_t< T1, T2, T_t0, T_ts > > >	integrate_ode_rk45 (const F &f, const std::vector< T1 > &y0, const T_t0 &t0, const std::vector< T_ts > &ts, const std::vector< T2 > &theta, const std::vector< double > &x, const std::vector< int > &x_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-6, double absolute_tolerance=1e-6, int max_num_steps=1E6)
	Return the solutions for the specified system of ordinary differential equations given the specified initial state, initial times, times of desired solution, and parameters and data, writing error and warning messages to the specified stream. More...
std::vector< int >	mpi_map_chunks (std::size_t num_jobs, std::size_t chunk_size=1)
	Maps jobs of given chunk size to workers and returning a vector of counts. More...
template<typename T >
std::unique_lock< std::mutex >	mpi_broadcast_command ()
	Broadcasts default constructible commands to the cluster. More...
std::unique_lock< std::mutex >	mpi_broadcast_command (std::shared_ptr< mpi_command > &command)
	Broadcasts a command instance to the listening cluster. More...
tbb::task_scheduler_init &	init_threadpool_tbb (tbb::stack_size_type stack_size=0)
	Initialize the Intel TBB threadpool and global scheduler through the tbb::task_scheduler_init object. More...
template<typename T_y >
void	check_cholesky_factor (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is a valid Cholesky factor. More...
template<typename T_y >
void	check_cholesky_factor_corr (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is a valid Cholesky factor of a correlation matrix. More...
template<typename T_y , int R, int C>
void	check_column_index (const char *function, const char *name, const Eigen::Matrix< T_y, R, C > &y, size_t i)
	Check if the specified index is a valid column of the matrix. More...
template<typename T >
void	check_consistent_size_mvt (const char *function, const char *name, const T &x, size_t expected_size)
	Check if the dimension of x is consistent, which is defined to be expected_size if x is a vector of vectors or 1 if x is a single vector. More...
template<typename T1 , typename T2 >
void	check_consistent_sizes_mvt (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2)
	Check if the dimension of x1 is consistent with x2. More...
template<typename T_y >
void	check_corr_matrix (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is a valid correlation matrix. More...
template<typename T_y >
void	check_cov_matrix (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is a valid covariance matrix. More...
void	check_flag_sundials (int flag, const char *func_name)
	Throws an exception when a Sundial function fails (i.e. More...
void	check_flag_kinsol (int flag, long int max_num_steps)
	Throws an exception message when the function KINSol() (call to the solver) fails. More...
template<typename T , int R, int C>
void	check_ldlt_factor (const char *function, const char *name, LDLT_factor< T, R, C > &A)
	Raise domain error if the specified LDLT factor is invalid. More...
template<typename T_y >
void	check_lower_triangular (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is lower triangular. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
void	check_matching_dims (const char *function, const char *name1, const Eigen::Matrix< T1, R1, C1 > &y1, const char *name2, const Eigen::Matrix< T2, R2, C2 > &y2)
	Check if the two matrices are of the same size. More...
template<bool check_compile, typename T1 , typename T2 , int R1, int C1, int R2, int C2>
void	check_matching_dims (const char *function, const char *name1, const Eigen::Matrix< T1, R1, C1 > &y1, const char *name2, const Eigen::Matrix< T2, R2, C2 > &y2)
	Check if the two matrices are of the same size. More...
template<typename T1 , typename T2 >
void	check_multiplicable (const char *function, const char *name1, const T1 &y1, const char *name2, const T2 &y2)
	Check if the matrices can be multiplied. More...
template<typename T_y >
void	check_nonempty (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is not empty. More...
template<typename T_y >
void	check_ordered (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, 1 > &y)
	Check if the specified vector is sorted into strictly increasing order. More...
template<typename T_y >
void	check_pos_definite (const char *function, const char *name, const Eigen::Matrix< T_y,-1,-1 > &y)
	Check if the specified square, symmetric matrix is positive definite. More...
template<typename Derived >
void	check_pos_definite (const char *function, const char *name, const Eigen::LDLT< Derived > &cholesky)
	Check if the specified LDLT transform of a matrix is positive definite. More...
template<typename Derived >
void	check_pos_definite (const char *function, const char *name, const Eigen::LLT< Derived > &cholesky)
	Check if the specified LLT decomposition transform resulted in Eigen::Success More...
template<typename T_y >
void	check_pos_semidefinite (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is positive definite. More...
template<typename Derived >
void	check_pos_semidefinite (const char *function, const char *name, const Eigen::LDLT< Derived > &cholesky)
	Check if the specified matrix is positive semidefinite. More...
template<typename T_y >
void	check_positive_ordered (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, 1 > &y)
	Check if the specified vector contains non-negative values and is sorted into strictly increasing order. More...
void	check_range (const char *function, const char *name, int max, int index, int nested_level, const char *error_msg)
	Check if specified index is within range. More...
void	check_range (const char *function, const char *name, int max, int index, const char *error_msg)
	Check if specified index is within range. More...
void	check_range (const char *function, const char *name, int max, int index)
	Check if specified index is within range. More...
template<typename T_y , int R, int C>
void	check_row_index (const char *function, const char *name, const Eigen::Matrix< T_y, R, C > &y, size_t i)
	Check if the specified index is a valid row of the matrix This check is 1-indexed by default. More...
template<typename T_prob >
void	check_simplex (const char *function, const char *name, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
	Check if the specified vector is simplex. More...
template<typename T_y >
void	check_square (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is square. More...
template<typename T >
void	check_std_vector_index (const char *function, const char *name, const std::vector< T > &y, int i)
	Check if the specified index is valid in std vector This check is 1-indexed by default. More...
template<typename T_y >
void	check_symmetric (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Check if the specified matrix is symmetric. More...
template<typename T_prob >
void	check_unit_vector (const char *function, const char *name, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
	Check if the specified vector is unit vector. More...
template<typename T , int R, int C>
void	check_vector (const char *function, const char *name, const Eigen::Matrix< T, R, C > &x)
	Check if the matrix is either a row vector or column vector. More...
template<typename T_y >
bool	is_cholesky_factor (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return true if y is a valid Choleksy factor, if number of rows is not less than the number of columns, if there are no 0 columns, and no element in matrix is NaN. More...
template<typename T_y >
bool	is_cholesky_factor_corr (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return true if y is a valid Cholesky factor, if the number of rows is not less than the number of columns, if there are no zero columns, and no element in matrix is NaN. More...
template<typename T_y , int R, int C>
bool	is_column_index (const Eigen::Matrix< T_y, R, C > &y, size_t i)
	Return true if column index is in bounds. More...
template<typename T_y >
bool	is_corr_matrix (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return true if the matrix is square and not 0x0, if the matrix is symmetric, diagonals are near 1, positive definite, and no elements are NaN A valid correlation matrix is symmetric, has a unit diagonal (all 1 values), and has all values between -1 and 1 (inclusive). More...
template<typename T , int R, int C>
bool	is_ldlt_factor (LDLT_factor< T, R, C > &A)
	Return true if the specified LDLT factor is invalid. More...
template<typename T_y >
bool	is_lower_triangular (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return true is matrix is lower triangular. More...
template<typename T , int R, int C>
bool	is_mat_finite (const Eigen::Matrix< T, R, C > &y)
	Return true is the specified matrix is finite. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
bool	is_matching_dims (const Eigen::Matrix< T1, R1, C1 > &y1, const Eigen::Matrix< T2, R2, C2 > &y2)
	Return true if the two matrices are of the same size. More...
template<bool check_compile, typename T1 , typename T2 , int R1, int C1, int R2, int C2>
bool	is_matching_dims (const Eigen::Matrix< T1, R1, C1 > &y1, const Eigen::Matrix< T2, R2, C2 > &y2)
	Return true if the two matrices are of the same size. More...
template<typename T_y >
bool	is_pos_definite (const Eigen::Matrix< T_y,-1,-1 > &y)
	Return true if the matrix is square or if the matrix has non-zero size, or if the matrix is symmetric, or if it is positive definite, or if no element is NaN. More...
template<typename Derived >
bool	is_pos_definite (const Eigen::LDLT< Derived > &cholesky)
	Return true if the matrix is positive definite. More...
template<typename Derived >
bool	is_pos_definite (const Eigen::LLT< Derived > &cholesky)
	Return true if diagonal of the L matrix is positive. More...
template<typename T_y >
bool	is_square (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return true if the matrix is square. More...
template<typename T_y >
bool	is_symmetric (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return true if the matrix is square, and no element not on the main diagonal is NaN. More...
template<typename T_prob >
bool	is_unit_vector (const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
	Return true if the vector is not a unit vector or if any element is NaN. More...
void	validate_non_negative_index (const char *var_name, const char *expr, int val)
template<typename T >
apply_scalar_unary< acos_fun, T >::return_t	acos (const T &x)
	Vectorized version of acos(). More...
template<typename T >
apply_scalar_unary< acosh_fun, T >::return_t	acosh (const T &x)
	Return the elementwise application of acosh() to specified argument container. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	add (const Eigen::Matrix< T1, R, C > &m1, const Eigen::Matrix< T2, R, C > &m2)
	Return the sum of the specified matrices. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	add (const Eigen::Matrix< T1, R, C > &m, const T2 &c)
	Return the sum of the specified matrix and specified scalar. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	add (const T1 &c, const Eigen::Matrix< T2, R, C > &m)
	Return the sum of the specified scalar and specified matrix. More...
template<typename T_m , typename T_a >
Eigen::Matrix< return_type_t< T_m, T_a >, Eigen::Dynamic, Eigen::Dynamic >	add_diag (const Eigen::Matrix< T_m, Eigen::Dynamic, Eigen::Dynamic > &mat, const T_a &to_add)
	Returns a Matrix with values added along the main diagonal. More...
template<typename T_m , typename T_a , int R, int C>
Eigen::Matrix< return_type_t< T_m, T_a >, Eigen::Dynamic, Eigen::Dynamic >	add_diag (const Eigen::Matrix< T_m, Eigen::Dynamic, Eigen::Dynamic > &mat, const Eigen::Matrix< T_a, R, C > &to_add)
	Returns a Matrix with values added along the main diagonal. More...
template<typename T1 , typename T2 >
append_return_type< std::vector< T1 >, std::vector< T2 > >::type	append_array (const std::vector< T1 > &x, const std::vector< T2 > &y)
	Return the concatenation of two specified vectors in the order of the arguments. More...
template<typename T1 >
std::vector< T1 >	append_array (const std::vector< T1 > &x, const std::vector< T1 > &y)
	Return the concatenation of two specified vectors in the order of the arguments. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, Eigen::Dynamic, Eigen::Dynamic >	append_col (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &B)
	Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix. More...
template<typename T1 , typename T2 , int C1, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, 1, Eigen::Dynamic >	append_col (const Eigen::Matrix< T1, 1, C1 > &A, const Eigen::Matrix< T2, 1, C2 > &B)
	Return the result of concatenating the first row vector followed by the second row vector side by side, with the result being a row vector. More...
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	append_col (const Eigen::Matrix< T, R1, C1 > &A, const Eigen::Matrix< T, R2, C2 > &B)
	Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix. More...
template<typename T , int C1, int C2>
Eigen::Matrix< T, 1, Eigen::Dynamic >	append_col (const Eigen::Matrix< T, 1, C1 > &A, const Eigen::Matrix< T, 1, C2 > &B)
	Return the result of concatenating the first row vector followed by the second row vector side by side, with the result being a row vector. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, 1, Eigen::Dynamic >	append_col (const T1 &A, const Eigen::Matrix< T2, R, C > &B)
	Return the result of stacking an scalar on top of the a row vector, with the result being a row vector. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, 1, Eigen::Dynamic >	append_col (const Eigen::Matrix< T1, R, C > &A, const T2 &B)
	Return the result of stacking a row vector on top of the an scalar, with the result being a row vector. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, Eigen::Dynamic, Eigen::Dynamic >	append_row (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &B)
	Return the result of stacking the rows of the first argument matrix on top of the second argument matrix. More...
template<typename T1 , typename T2 , int R1, int R2>
Eigen::Matrix< return_type_t< T1, T2 >, Eigen::Dynamic, 1 >	append_row (const Eigen::Matrix< T1, R1, 1 > &A, const Eigen::Matrix< T2, R2, 1 > &B)
	Return the result of stacking the first vector on top of the second vector, with the result being a vector. More...
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	append_row (const Eigen::Matrix< T, R1, C1 > &A, const Eigen::Matrix< T, R2, C2 > &B)
	Return the result of stacking the rows of the first argument matrix on top of the second argument matrix. More...
template<typename T , int R1, int R2>
Eigen::Matrix< T, Eigen::Dynamic, 1 >	append_row (const Eigen::Matrix< T, R1, 1 > &A, const Eigen::Matrix< T, R2, 1 > &B)
	Return the result of stacking the first vector on top of the second vector, with the result being a vector. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, Eigen::Dynamic, 1 >	append_row (const T1 &A, const Eigen::Matrix< T2, R, C > &B)
	Return the result of stacking an scalar on top of the a vector, with the result being a vector. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, Eigen::Dynamic, 1 >	append_row (const Eigen::Matrix< T1, R, C > &A, const T2 &B)
	Return the result of stacking a vector on top of the an scalar, with the result being a vector. More...
template<typename T >
apply_scalar_unary< asin_fun, T >::return_t	asin (const T &x)
	Vectorized version of asin(). More...
template<typename T >
apply_scalar_unary< asinh_fun, T >::return_t	asinh (const T &x)
	Vectorized version of asinh(). More...
template<int N>
void	print_mat_size (std::ostream &o)
	Helper function to return the matrix size as either "dynamic" or "1". More...
template<typename T_lhs , typename T_rhs >
void	assign (T_lhs &x, const T_rhs &y)
	Copy the right-hand side's value to the left-hand side variable. More...
template<typename T_lhs , typename T_rhs , int R1, int C1, int R2, int C2>
void	assign (Eigen::Matrix< T_lhs, R1, C1 > &x, const Eigen::Matrix< T_rhs, R2, C2 > &y)
	Copy the right-hand side's value to the left-hand side variable. More...
template<typename T_lhs , typename T_rhs , int R, int C>
void	assign (Eigen::Matrix< T_lhs, R, C > &x, const Eigen::Matrix< T_rhs, R, C > &y)
	Copy the right-hand side's value to the left-hand side variable. More...
template<typename T_lhs , typename T , int R, int C>
void	assign (Eigen::Block< T_lhs > x, const Eigen::Matrix< T, R, C > &y)
	Copy the right-hand side's value to the left-hand side variable. More...
template<typename T_lhs , typename T_rhs >
void	assign (std::vector< T_lhs > &x, const std::vector< T_rhs > &y)
	Copy the right-hand side's value to the left-hand side variable. More...
template<typename T >
apply_scalar_unary< atan_fun, T >::return_t	atan (const T &x)
	Vectorized version of atan(). More...
template<typename T >
apply_scalar_unary< atanh_fun, T >::return_t	atanh (const T &x)
	Return the elementwise application of atanh() to specified argument container. More...
template<typename T >
void	autocorrelation (const std::vector< T > &y, std::vector< T > &ac, Eigen::FFT< T > &fft)
	Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
template<typename T , typename DerivedA , typename DerivedB >
void	autocorrelation (const Eigen::MatrixBase< DerivedA > &y, Eigen::MatrixBase< DerivedB > &ac, Eigen::FFT< T > &fft)
	Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
template<typename T >
void	autocorrelation (const std::vector< T > &y, std::vector< T > &ac)
	Write autocorrelation estimates for every lag for the specified input sequence into the specified result. More...
template<typename T , typename DerivedA , typename DerivedB >
void	autocorrelation (const Eigen::MatrixBase< DerivedA > &y, Eigen::MatrixBase< DerivedB > &ac)
	Write autocorrelation estimates for every lag for the specified input sequence into the specified result. More...
template<typename T >
void	autocovariance (const std::vector< T > &y, std::vector< T > &acov, Eigen::FFT< T > &fft)
	Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
template<typename T , typename DerivedA , typename DerivedB >
void	autocovariance (const Eigen::MatrixBase< DerivedA > &y, Eigen::MatrixBase< DerivedB > &acov, Eigen::FFT< T > &fft)
	Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
template<typename T >
void	autocovariance (const std::vector< T > &y, std::vector< T > &acov)
	Write autocovariance estimates for every lag for the specified input sequence into the specified result. More...
template<typename T , typename DerivedA , typename DerivedB >
void	autocovariance (const Eigen::MatrixBase< DerivedA > &y, Eigen::MatrixBase< DerivedB > &acov)
	Write autocovariance estimates for every lag for the specified input sequence into the specified result. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	block (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t i, size_t j, size_t nrows, size_t ncols)
	Return a nrows x ncols submatrix starting at (i-1, j-1). More...
template<typename T >
apply_scalar_unary< cbrt_fun, T >::return_t	cbrt (const T &x)
	Vectorized version of cbrt(). More...
template<typename T >
apply_scalar_unary< ceil_fun, T >::return_t	ceil (const T &x)
	Vectorized version of ceil(). More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	chol2inv (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &L)
	Returns the inverse of the matrix whose Cholesky factor is L. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cholesky_corr_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y, int K)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cholesky_corr_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y, int K, T &lp)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	cholesky_corr_free (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cholesky_decompose (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Return the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix. More...
template<>
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >	cholesky_decompose (const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &m)
	Return the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cholesky_factor_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, int M, int N)
	Return the Cholesky factor of the specified size read from the specified vector. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cholesky_factor_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, int M, int N, T &lp)
	Return the Cholesky factor of the specified size read from the specified vector and increment the specified log probability reference with the log Jacobian adjustment of the transform. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	cholesky_factor_free (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return the unconstrained vector of parameters correspdonding to the specified Cholesky factor. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	col (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t j)
	Return the specified column of the specified matrix using start-at-1 indexing. More...
template<typename T , int R, int C>
int	cols (const Eigen::Matrix< T, R, C > &m)
	Return the number of columns in the specified matrix, vector, or row vector. More...
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< double, 1, C1 >	columns_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
	Returns the dot product of the specified vectors. More...
template<typename T , int R, int C>
Eigen::Matrix< T, 1, C >	columns_dot_self (const Eigen::Matrix< T, R, C > &x)
	Returns the dot product of each column of a matrix with itself. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	corr_matrix_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 >>::type k)
	Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	corr_matrix_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 >>::type k, T &lp)
	Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	corr_matrix_free (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return the vector of unconstrained partial correlations that define the specified correlation matrix when transformed. More...
template<typename T >
apply_scalar_unary< cos_fun, T >::return_t	cos (const T &x)
	Vectorized version of cos(). More...
template<typename T >
apply_scalar_unary< cosh_fun, T >::return_t	cosh (const T &x)
	Vectorized version of cosh(). More...
template<typename T_x , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	cov_exp_quad (const std::vector< T_x > &x, const T_sigma &sigma, const T_l &length_scale)
template<typename T_x , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	cov_exp_quad (const std::vector< T_x > &x, const T_sigma &sigma, const std::vector< T_l > &length_scale)
template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	cov_exp_quad (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma, const T_l &length_scale)
template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	cov_exp_quad (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma, const std::vector< T_l > &length_scale)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cov_matrix_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 >>::type K)
	Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cov_matrix_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >>::type K, T &lp)
	Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cov_matrix_constrain_lkj (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t k)
	Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	cov_matrix_constrain_lkj (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t k, T &lp)
	Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values and increment the specified log probability reference with the log absolute Jacobian determinant. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	cov_matrix_free (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &y)
	The covariance matrix derived from the symmetric view of the lower-triangular view of the K by K specified matrix is freed to return a vector of size K + (K choose 2). More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	cov_matrix_free_lkj (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &y)
	Return the vector of unconstrained partial correlations and deviations that transform to the specified covariance matrix. More...
matrix_d	crossprod (const matrix_d &M)
	Returns the result of pre-multiplying a matrix by its own transpose. More...
template<typename T >
const std::vector< int >	csr_extract_u (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
	Extract the NZE index for each entry from a sparse matrix. More...
template<typename T , int R, int C>
const std::vector< int >	csr_extract_u (const Eigen::Matrix< T, R, C > &A)
	Extract the NZE index for each entry from a sparse matrix. More...
template<typename T >
const std::vector< int >	csr_extract_v (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
	Extract the column indexes for non-zero value from a sparse matrix. More...
template<typename T , int R, int C>
const std::vector< int >	csr_extract_v (const Eigen::Matrix< T, R, C > &A)
	Extract the column indexes for non-zero values from a dense matrix by converting to sparse and calling the sparse matrix extractor. More...
template<typename T >
const Eigen::Matrix< T, Eigen::Dynamic, 1 >	csr_extract_w (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
template<typename T , int R, int C>
const Eigen::Matrix< T, Eigen::Dynamic, 1 >	csr_extract_w (const Eigen::Matrix< T, R, C > &A)
template<typename T1 , typename T2 >
Eigen::Matrix< return_type_t< T1, T2 >, Eigen::Dynamic, 1 >	csr_matrix_times_vector (int m, int n, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &w, const std::vector< int > &v, const std::vector< int > &u, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &b)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	csr_to_dense_matrix (int m, int n, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &w, const std::vector< int > &v, const std::vector< int > &u)
	Construct a dense Eigen matrix from the CSR format components. More...
int	csr_u_to_z (const std::vector< int > &u, int i)
	Return the z vector computed from the specified u vector at the index for the z vector. More...
template<typename T >
std::vector< T >	cumulative_sum (const std::vector< T > &x)
	Return the cumulative sum of the specified vector. More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	cumulative_sum (const Eigen::Matrix< T, R, C > &m)
	Return the cumulative sum of the specified matrix. More...
template<typename T , int R, int C>
T	determinant (const Eigen::Matrix< T, R, C > &m)
	Returns the determinant of the specified square matrix. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	diag_matrix (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v)
	Return a square diagonal matrix with the specified vector of coefficients as the diagonal values. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C1 >	diag_post_multiply (const Eigen::Matrix< T1, R1, C1 > &m1, const Eigen::Matrix< T2, R2, C2 > &m2)
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R2, C2 >	diag_pre_multiply (const Eigen::Matrix< T1, R1, C1 > &m1, const Eigen::Matrix< T2, R2, C2 > &m2)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	diagonal (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Return a column vector of the diagonal elements of the specified matrix. More...
template<typename T >
apply_scalar_unary< digamma_fun, T >::return_t	digamma (const T &x)
	Vectorized version of digamma(). More...
template<typename T >
void	dims (const T &x, std::vector< int > &result)
template<typename T , int R, int C>
void	dims (const Eigen::Matrix< T, R, C > &x, std::vector< int > &result)
template<typename T >
void	dims (const std::vector< T > &x, std::vector< int > &result)
template<typename T >
std::vector< int >	dims (const T &x)
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
return_type_t< T1, T2 >	distance (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
	Returns the distance between the specified vectors. More...
template<int R, int C, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	divide (const Eigen::Matrix< T1, R, C > &m, T2 c)
	Return matrix divided by scalar. More...
template<typename T_x , typename T_v >
std::vector< Eigen::Matrix< return_type_t< T_x, T_v, double >, Eigen::Dynamic, 1 > >	divide_columns (const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &x, const std::vector< T_v > &vec)
	Takes Stan data type vector[n] x[D] and divides column vector in x element-wise by the values in vec. More...
template<int R1, int C1, int R2, int C2>
double	dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
	Returns the dot product of the specified vectors. More...
double	dot_product (const double *v1, const double *v2, size_t length)
	Returns the dot product of the specified arrays of doubles. More...
double	dot_product (const std::vector< double > &v1, const std::vector< double > &v2)
	Returns the dot product of the specified arrays of doubles. More...
template<int R, int C>
double	dot_self (const Eigen::Matrix< double, R, C > &v)
	Returns the dot product of the specified vector with itself. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	eigenvalues_sym (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Return the eigenvalues of the specified symmetric matrix in descending order of magnitude. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	eigenvectors_sym (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	elt_divide (const Eigen::Matrix< T1, R, C > &m1, const Eigen::Matrix< T2, R, C > &m2)
	Return the elementwise division of the specified matrices. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	elt_divide (const Eigen::Matrix< T1, R, C > &m, T2 s)
	Return the elementwise division of the specified matrix by the specified scalar. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	elt_divide (T1 s, const Eigen::Matrix< T2, R, C > &m)
	Return the elementwise division of the specified scalar by the specified matrix. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	elt_multiply (const Eigen::Matrix< T1, R, C > &m1, const Eigen::Matrix< T2, R, C > &m2)
	Return the elementwise multiplication of the specified matrices. More...
template<typename T >
apply_scalar_unary< erf_fun, T >::return_t	erf (const T &x)
	Vectorized version of erf(). More...
template<typename T >
apply_scalar_unary< erfc_fun, T >::return_t	erfc (const T &x)
	Vectorized version of erfc(). More...
template<typename T >
apply_scalar_unary< exp_fun, T >::return_t	exp (const T &x)
	Return the elementwise exponentiation of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<typename T , typename = require_vector_like_t<T>>
apply_scalar_unary< exp2_fun, T >::return_t	exp2 (const T &x)
	Return the elementwise application of exp2() to specified argument container. More...
template<typename T >
apply_scalar_unary< expm1_fun, T >::return_t	expm1 (const T &x)
	Vectorized version of expm1(). More...
template<typename T >
apply_scalar_unary< fabs_fun, T >::return_t	fabs (const T &x)
	Vectorized version of fabs(). More...
template<typename T >
bool	factor_cov_matrix (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &Sigma, Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, Eigen::Array< T, Eigen::Dynamic, 1 > &sds)
	This function is intended to make starting values, given a covariance matrix Sigma. More...
template<typename T >
void	factor_U (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &U, Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs)
	This function is intended to make starting values, given a unit upper-triangular matrix U such that U'DU is a correlation matrix. More...
template<typename T , int R, int C, typename S >
void	fill (Eigen::Matrix< T, R, C > &x, const S &y)
	Fill the specified container with the specified value. More...
template<typename T >
apply_scalar_unary< floor_fun, T >::return_t	floor (const T &x)
	Vectorized version of floor(). More...
template<typename T >
const T &	get_base1 (const std::vector< T > &x, size_t i, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one index. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< T > > &x, size_t i1, size_t i2, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< T > > > &x, size_t i1, size_t i2, size_t i3, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< T > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
const T &	get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, size_t i8, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic >	get_base1 (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, const char *error_msg, size_t idx)
	Return a copy of the row of the specified vector at the specified base-one row index. More...
template<typename T >
const T &	get_base1 (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, size_t n, const char *error_msg, size_t idx)
	Return a reference to the value of the specified matrix at the specified base-one row and column indexes. More...
template<typename T >
const T &	get_base1 (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t m, const char *error_msg, size_t idx)
	Return a reference to the value of the specified column vector at the specified base-one index. More...
template<typename T >
const T &	get_base1 (const Eigen::Matrix< T, 1, Eigen::Dynamic > &x, size_t n, const char *error_msg, size_t idx)
	Return a reference to the value of the specified row vector at the specified base-one index. More...
template<typename T >
T &	get_base1_lhs (std::vector< T > &x, size_t i, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one index. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< T > > &x, size_t i1, size_t i2, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< T > > > &x, size_t i1, size_t i2, size_t i3, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< T > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
T &	get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, size_t i8, const char *error_msg, size_t idx)
	Return a reference to the value of the specified vector at the specified base-one indexes. More...
template<typename T >
Eigen::Block< Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > >	get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, const char *error_msg, size_t idx)
	Return a copy of the row of the specified vector at the specified base-one row index. More...
template<typename T >
T &	get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, size_t n, const char *error_msg, size_t idx)
	Return a reference to the value of the specified matrix at the specified base-one row and column indexes. More...
template<typename T >
T &	get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t m, const char *error_msg, size_t idx)
	Return a reference to the value of the specified column vector at the specified base-one index. More...
template<typename T >
T &	get_base1_lhs (Eigen::Matrix< T, 1, Eigen::Dynamic > &x, size_t n, const char *error_msg, size_t idx)
	Return a reference to the value of the specified row vector at the specified base-one index. More...
template<typename T_lp , typename T_lp_accum >
return_type_t< T_lp, T_lp_accum >	get_lp (const T_lp &lp, const accumulator< T_lp_accum > &lp_accum)
template<typename T_x , typename T_sigma >
Eigen::Matrix< return_type_t< T_x, T_sigma >, Eigen::Dynamic, Eigen::Dynamic >	gp_dot_prod_cov (const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &x, const T_sigma &sigma)
	Returns a dot product covariance matrix. More...
template<typename T_x , typename T_sigma >
Eigen::Matrix< return_type_t< T_x, T_sigma >, Eigen::Dynamic, Eigen::Dynamic >	gp_dot_prod_cov (const std::vector< T_x > &x, const T_sigma &sigma)
	Returns a dot product covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_sigma >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma >, Eigen::Dynamic, Eigen::Dynamic >	gp_dot_prod_cov (const std::vector< Eigen::Matrix< T_x1, Eigen::Dynamic, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2, Eigen::Dynamic, 1 >> &x2, const T_sigma &sigma)
	Returns a dot product covariance matrix of differing x's. More...
template<typename T_x1 , typename T_x2 , typename T_sigma >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma >, Eigen::Dynamic, Eigen::Dynamic >	gp_dot_prod_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma)
	Returns a dot product covariance matrix of differing x's. More...
template<typename T_x , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exp_quad_cov (const std::vector< T_x > &x, const T_sigma &sigma, const T_l &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exp_quad_cov (const std::vector< Eigen::Matrix< T_x,-1, 1 >> &x, const T_sigma &sigma, const std::vector< T_l > &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exp_quad_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma, const T_l &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exp_quad_cov (const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &x2, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a squared exponential kernel. More...
template<>
Eigen::MatrixXd	gp_exp_quad_cov (const std::vector< double > &x, const double &sigma, const double &length_scale)
	Returns a squared exponential kernel. More...
template<>
Eigen::MatrixXd	gp_exp_quad_cov (const std::vector< Eigen::VectorXd > &x, const double &sigma, const double &length_scale)
	Returns a squared exponential kernel. More...
template<>
Eigen::MatrixXd	gp_exp_quad_cov (const std::vector< Eigen::VectorXd > &x, const double &sigma, const std::vector< double > &length_scale)
	Returns a squared exponential kernel. More...
template<>
Eigen::MatrixXd	gp_exp_quad_cov (const std::vector< double > &x1, const std::vector< double > &x2, const double &sigma, const double &length_scale)
	Returns a squared exponential kernel. More...
template<>
Eigen::MatrixXd	gp_exp_quad_cov (const std::vector< Eigen::VectorXd > &x1, const std::vector< Eigen::VectorXd > &x2, const double &sigma, const double &length_scale)
	Returns a squared exponential kernel. More...
template<>
Eigen::MatrixXd	gp_exp_quad_cov (const std::vector< Eigen::VectorXd > &x1, const std::vector< Eigen::VectorXd > &x2, const double &sigma, const std::vector< double > &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exponential_cov (const std::vector< T_x > &x, const T_s &sigma, const T_l &length_scale)
	Returns a Matern exponential covariance Matrix. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exponential_cov (const std::vector< Eigen::Matrix< T_x,-1, 1 >> &x, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern exponential covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exponential_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_s &sigma, const T_l &length_scale)
	Returns a Matern exponential cross covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_exponential_cov (const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &x2, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern exponential cross covariance matrix. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern32_cov (const std::vector< T_x > &x, const T_s &sigma, const T_l &length_scale)
	Returns a Matern 3/2 covariance matrix. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern32_cov (const std::vector< Eigen::Matrix< T_x,-1, 1 >> &x, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern 3/2 covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern32_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_s &sigma, const T_l &length_scale)
	Returns a Matern 3/2 cross covariance matrix. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern32_cov (const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &x2, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern 3/2 cross covariance matrix. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern52_cov (const std::vector< T_x > &x, const T_s &sigma, const T_l &length_scale)
	Returns a Matern 5/2 covariance matrix with one input vector. More...
template<typename T_x , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern52_cov (const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &x, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern 5/2 covariance matrix with one input vector with automatic relevance determination (ARD). More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern52_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_s &sigma, const T_l &length_scale)
	Returns a Matern 5/2 covariance matrix with two different input vectors. More...
template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_s, T_l >, Eigen::Dynamic, Eigen::Dynamic >	gp_matern52_cov (const std::vector< Eigen::Matrix< T_x1, Eigen::Dynamic, 1 >> &x1, const std::vector< Eigen::Matrix< T_x2, Eigen::Dynamic, 1 >> &x2, const T_s &sigma, const std::vector< T_l > &length_scale)
	Returns a Matern 5/2 covariance matrix with two input vectors with automatic relevance determination (ARD). More...
template<typename T_x , typename T_sigma , typename T_l , typename T_p >
Eigen::Matrix< return_type_t< T_x, T_sigma, T_l, T_p >, Eigen::Dynamic, Eigen::Dynamic >	gp_periodic_cov (const std::vector< T_x > &x, const T_sigma &sigma, const T_l &l, const T_p &p)
	Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \). More...
template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l , typename T_p >
Eigen::Matrix< return_type_t< T_x1, T_x2, T_sigma, T_l, T_p >, Eigen::Dynamic, Eigen::Dynamic >	gp_periodic_cov (const std::vector< T_x1 > &x1, const std::vector< T_x2 > &x2, const T_sigma &sigma, const T_l &l, const T_p &p)
	Returns a periodic covariance matrix \( \mathbf{K} \) using inputs \( \mathbf{X}_1 \) and \( \mathbf{X}_2 \). More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	head (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, size_t n)
	Return the specified number of elements as a vector from the front of the specified vector. More...
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic >	head (const Eigen::Matrix< T, 1, Eigen::Dynamic > &rv, size_t n)
	Return the specified number of elements as a row vector from the front of the specified row vector. More...
template<typename T >
std::vector< T >	head (const std::vector< T > &sv, size_t n)
	Return the specified number of elements as a standard vector from the front of the specified standard vector. More...
template<typename T >
void	initialize (T &x, const T &v)
template<typename T , typename V , typename = require_arithmetic_t<V>>
void	initialize (T &x, V v)
template<typename T , int R, int C, typename V >
void	initialize (Eigen::Matrix< T, R, C > &x, const V &v)
template<typename T , typename V >
void	initialize (std::vector< T > &x, const V &v)
template<typename T >
apply_scalar_unary< inv_fun, T >::return_t	inv (const T &x)
	Vectorized version of inv(). More...
template<typename T >
apply_scalar_unary< inv_cloglog_fun, T >::return_t	inv_cloglog (const T &x)
	Vectorized version of inv_cloglog(). More...
template<typename T >
apply_scalar_unary< inv_logit_fun, T >::return_t	inv_logit (const T &x)
	Vectorized version of inv_logit(). More...
template<typename T >
apply_scalar_unary< inv_Phi_fun, T >::return_t	inv_Phi (const T &x)
	Vectorized version of inv_Phi(). More...
template<typename T >
apply_scalar_unary< inv_sqrt_fun, T >::return_t	inv_sqrt (const T &x)
	Vectorized version of inv_sqrt(). More...
template<typename T >
apply_scalar_unary< inv_square_fun, T >::return_t	inv_square (const T &x)
	Vectorized version of inv_square(). More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	inverse (const Eigen::Matrix< T, R, C > &m)
	Returns the inverse of the specified matrix. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	inverse_spd (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Returns the inverse of the specified symmetric, pos/neg-definite matrix. More...
template<typename T >
apply_scalar_unary< lgamma_fun, T >::return_t	lgamma (const T &x)
	Vectorized version of lgamma(). More...
template<typename T >
apply_scalar_unary< log_fun, T >::return_t	log (const T &x)
	Return the elementwise natural log of the specified argument, which may be a scalar or any Stan container of numeric scalars. More...
template<typename T >
apply_scalar_unary< log10_fun, T >::return_t	log10 (const T &x)
	Vectorized version of log10(). More...
template<typename T >
apply_scalar_unary< log1m_fun, T >::return_t	log1m (const T &x)
	Vectorized version of log1m(). More...
template<typename T >
apply_scalar_unary< log1m_exp_fun, T >::return_t	log1m_exp (const T &x)
	Vectorized version of log1m_exp(). More...
template<typename T >
apply_scalar_unary< log1m_inv_logit_fun, T >::return_t	log1m_inv_logit (const T &x)
	Return the elementwise application of log1m_inv_logit() to specified argument container. More...
template<typename T >
apply_scalar_unary< log1p_fun, T >::return_t	log1p (const T &x)
	Return the elementwise application of log1p() to specified argument container. More...
template<typename T >
apply_scalar_unary< log1p_exp_fun, T >::return_t	log1p_exp (const T &x)
	Vectorized version of log1m_exp(). More...
template<typename T , typename = require_vector_like_t<T>>
apply_scalar_unary< log2_fun, T >::return_t	log2 (const T &x)
	Return the elementwise application of log2() to specified argument container. More...
template<typename T , int R, int C>
T	log_determinant (const Eigen::Matrix< T, R, C > &m)
	Returns the log absolute determinant of the specified square matrix. More...
template<int R, int C, typename T >
T	log_determinant_ldlt (LDLT_factor< T, R, C > &A)
	Returns log(abs(det(A))) given a LDLT_factor of A. More...
template<typename T , int R, int C>
T	log_determinant_spd (const Eigen::Matrix< T, R, C > &m)
	Returns the log absolute determinant of the specified square matrix. More...
template<typename T >
apply_scalar_unary< log_inv_logit_fun, T >::return_t	log_inv_logit (const T &x)
	Return the elementwise application of log_inv_logit() to specified argument container. More...
template<typename T_theta , typename T_lam >
return_type_t< T_theta, T_lam >	log_mix (const T_theta &theta, const T_lam &lambda)
	Return the log mixture density with specified mixing proportions and log densities. More...
template<typename T_theta , typename T_lam , int R, int C>
return_type_t< T_theta, std::vector< Eigen::Matrix< T_lam, R, C > > >	log_mix (const T_theta &theta, const std::vector< Eigen::Matrix< T_lam, R, C >> &lambda)
	Return the log mixture density given specified mixing proportions and array of log density vectors. More...
template<typename T_theta , typename T_lam >
return_type_t< T_theta, std::vector< std::vector< T_lam > > >	log_mix (const T_theta &theta, const std::vector< std::vector< T_lam >> &lambda)
	Return the log mixture density given specified mixing proportions and array of log density arrays. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	log_softmax (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v)
	Return the natural logarithm of the softmax of the specified vector. More...
template<int R, int C>
double	log_sum_exp (const Eigen::Matrix< double, R, C > &x)
	Return the log of the sum of the exponentiated values of the specified matrix of values. More...
template<typename T >
apply_scalar_unary< logit_fun, T >::return_t	logit (const T &x)
	Return the elementwise application of logit() to specified argument container. More...
template<typename T >
Eigen::Array< T, Eigen::Dynamic, 1 >	make_nu (const T &eta, size_t K)
	Return the degrees of freedom for the t distribution that corresponds to the shape parameter in the LKJ distribution. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	matrix_exp (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &A)
	Return the matrix exponential of the input matrix. More...
template<typename T , int N>
Eigen::Matrix< T, N, N >	matrix_exp (const Eigen::Matrix< T, N, N > &A)
	Return the matrix exponential of the input statically-sized square matrix. More...
template<typename T >
Eigen::Matrix< T, 1, 1 >	matrix_exp (const Eigen::Matrix< T, 1, 1 > &A)
	Return the exponential of the input scalar when it's in the form of Eigen matrix. More...
template<typename Mtype >
Mtype	matrix_exp_2x2 (const Mtype &A)
	Return the matrix exponential of a 2x2 matrix. More...
template<int Cb>
Eigen::Matrix< double,-1, Cb >	matrix_exp_multiply (const Eigen::MatrixXd &A, const Eigen::Matrix< double,-1, Cb > &B)
	Return product of exp(A) and B, where A is a NxN double matrix, B is a NxCb double matrix, and t is a double. More...
template<typename MatrixType >
MatrixType	matrix_exp_pade (const MatrixType &arg)
	Computes the matrix exponential, using a Pade approximation, coupled with scaling and squaring. More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	matrix_power (const Eigen::Matrix< T, R, C > &M, const int n)
	Returns the nth power of the specific matrix. More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	operator^ (const Eigen::Matrix< T, R, C > &M, const int n)
int	max (const std::vector< int > &x)
	Returns the maximum coefficient in the specified column vector. More...
template<typename T >
T	max (const std::vector< T > &x)
	Returns the maximum coefficient in the specified column vector. More...
template<typename T , int R, int C>
T	max (const Eigen::Matrix< T, R, C > &m)
	Returns the maximum coefficient in the specified matrix, vector, or row vector. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_left (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
	Returns the solution of the system Ax=b. More...
template<int R1, int C1, int R2, int C2, typename T1 , typename T2 >
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_left_ldlt (const LDLT_factor< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_left_spd (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
	Returns the solution of the system Ax=b where A is symmetric positive definite. More...
template<int TriView, typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_left_tri (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
	Returns the solution of the system Ax=b when A is triangular. More...
template<int TriView, typename T , int R1, int C1>
Eigen::Matrix< T, R1, C1 >	mdivide_left_tri (const Eigen::Matrix< T, R1, C1 > &A)
	Returns the solution of the system Ax=b when A is triangular and b=I. More...
template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< double, R1, C2 >	mdivide_left_tri (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
	Returns the solution of the system Ax=b when A is triangular and A and b are matrices of doubles. More...
template<Eigen::UpLoType TriView, int R1, int C1>
Eigen::Matrix< double, R1, C1 >	mdivide_left_tri (const Eigen::Matrix< double, R1, C1 > &A)
	Returns the solution of the system Ax=b when A is triangular, b=I and both are matrices of doubles. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_left_tri_low (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
	Return the result of left dividing the second argument by the first argument. More...
template<typename T , int R1, int C1>
Eigen::Matrix< T, R1, C1 >	mdivide_left_tri_low (const Eigen::Matrix< T, R1, C1 > &A)
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_right (const Eigen::Matrix< T1, R1, C1 > &b, const Eigen::Matrix< T2, R2, C2 > &A)
	Returns the solution of the system xA=b. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_right_ldlt (const Eigen::Matrix< T1, R1, C1 > &b, const LDLT_factor< T2, R2, C2 > &A)
	Returns the solution of the system xA=b given an LDLT_factor of A. More...
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< double, R1, C2 >	mdivide_right_ldlt (const Eigen::Matrix< double, R1, C1 > &b, const LDLT_factor< double, R2, C2 > &A)
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_right_spd (const Eigen::Matrix< T1, R1, C1 > &b, const Eigen::Matrix< T2, R2, C2 > &A)
	Returns the solution of the system xA=b where A is symmetric positive definite. More...
template<Eigen::UpLoType TriView, typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_right_tri (const Eigen::Matrix< T1, R1, C1 > &b, const Eigen::Matrix< T2, R2, C2 > &A)
	Returns the solution of the system xA=b when A is triangular. More...
template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< double, R1, C2 >	mdivide_right_tri (const Eigen::Matrix< double, R1, C1 > &b, const Eigen::Matrix< double, R2, C2 > &A)
	Returns the solution of the system xA=b when A is triangular and A and b are matrices of doubles. More...
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	mdivide_right_tri_low (const Eigen::Matrix< T1, R1, C1 > &b, const Eigen::Matrix< T2, R2, C2 > &A)
	Returns the solution of the system x tri(A) = b when tri(A) is a lower triangular view of the matrix A. More...
template<typename T >
return_type_t< T >	mean (const std::vector< T > &v)
	Returns the sample mean (i.e., average) of the coefficients in the specified standard vector. More...
template<typename T , int R, int C>
return_type_t< T >	mean (const Eigen::Matrix< T, R, C > &m)
	Returns the sample mean (i.e., average) of the coefficients in the specified vector, row vector, or matrix. More...
int	min (const std::vector< int > &x)
	Returns the minimum coefficient in the specified column vector. More...
template<typename T >
T	min (const std::vector< T > &x)
	Returns the minimum coefficient in the specified column vector. More...
template<typename T , int R, int C>
T	min (const Eigen::Matrix< T, R, C > &m)
	Returns the minimum coefficient in the specified matrix, vector, or row vector. More...
template<typename T >
T	minus (const T &x)
	Returns the negation of the specified scalar or matrix. More...
template<int R, int C, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	multiply (const Eigen::Matrix< T1, R, C > &m, T2 c)
	Return specified matrix multiplied by specified scalar. More...
template<int R, int C, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	multiply (T1 c, const Eigen::Matrix< T2, R, C > &m)
	Return specified scalar multiplied by specified matrix. More...
template<int R1, int C1, int R2, int C2, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
Eigen::Matrix< return_type_t< T1, T2 >, R1, C2 >	multiply (const Eigen::Matrix< T1, R1, C1 > &m1, const Eigen::Matrix< T2, R2, C2 > &m2)
	Return the product of the specified matrices. More...
template<int C1, int R2, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
return_type_t< T1, T2 >	multiply (const Eigen::Matrix< T1, 1, C1 > &rv, const Eigen::Matrix< T2, R2, 1 > &v)
	Return the scalar product of the specified row vector and specified column vector. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
return_type_t< T1, T2 >	multiply (T1 m, T2 c)
	Return product of scalars. More...
matrix_d	multiply_lower_tri_self_transpose (const matrix_d &L)
	Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose. More...
template<typename T >
int	num_elements (const T &x)
	Returns 1, the number of elements in a primitive type. More...
template<typename T , int R, int C>
int	num_elements (const Eigen::Matrix< T, R, C > &m)
	Returns the size of the specified matrix. More...
template<typename T >
int	num_elements (const std::vector< T > &v)
	Returns the number of elements in the specified vector. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	ordered_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
	Return an increasing ordered vector derived from the specified free vector. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	ordered_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &lp)
	Return a positive valued, increasing ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	ordered_free (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y)
	Return the vector of unconstrained scalars that transform to the specified positive ordered vector. More...
template<typename T >
apply_scalar_unary< Phi_fun, T >::return_t	Phi (const T &x)
	Vectorized version of Phi(). More...
template<typename T >
apply_scalar_unary< Phi_approx_fun, T >::return_t	Phi_approx (const T &x)
	Return the elementwise application of Phi_approx() to specified argument container. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	positive_ordered_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
	Return an increasing positive ordered vector derived from the specified free vector. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	positive_ordered_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &lp)
	Return a positive valued, increasing positive ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	positive_ordered_free (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y)
	Return the vector of unconstrained scalars that transform to the specified positive ordered vector. More...
template<typename T >
T	prod (const std::vector< T > &v)
	Returns the product of the coefficients of the specified standard vector. More...
template<typename T , int R, int C>
T	prod (const Eigen::Matrix< T, R, C > &v)
	Returns the product of the coefficients of the specified column vector. More...
template<typename T1 , typename T2 , typename F >
common_type< T1, T2 >::type	promote_common (const F &u)
	Return the result of promoting either a scalar or the scalar elements of a container to either of two specified types, as determined by stan::math::common_type. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	qr_Q (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Returns the orthogonal factor of the fat QR decomposition. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	qr_R (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Returns the upper triangular factor of the fat QR decomposition. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	qr_thin_Q (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Returns the orthogonal factor of the thin QR decomposition. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	qr_thin_R (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Returns the upper triangular factor of the thin QR decomposition. More...
template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix< T, CB, CB >	quad_form (const Eigen::Matrix< T, RA, CA > &A, const Eigen::Matrix< T, RB, CB > &B)
	Compute B^T A B. More...
template<int RA, int CA, int RB, typename T >
T	quad_form (const Eigen::Matrix< T, RA, CA > &A, const Eigen::Matrix< T, RB, 1 > &B)
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, Eigen::Dynamic, Eigen::Dynamic >	quad_form_diag (const Eigen::Matrix< T1, Eigen::Dynamic, Eigen::Dynamic > &mat, const Eigen::Matrix< T2, R, C > &vec)
template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix< T, CB, CB >	quad_form_sym (const Eigen::Matrix< T, RA, CA > &A, const Eigen::Matrix< T, RB, CB > &B)
template<int RA, int CA, int RB, typename T >
T	quad_form_sym (const Eigen::Matrix< T, RA, CA > &A, const Eigen::Matrix< T, RB, 1 > &B)
template<typename C >
int	rank (const C &v, int s)
	Return the number of components of v less than v[s]. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	read_corr_L (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, size_t K)
	Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	read_corr_L (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, size_t K, T &log_prob)
	Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	read_corr_matrix (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, size_t K)
	Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	read_corr_matrix (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, size_t K, T &log_prob)
	Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	read_cov_L (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const Eigen::Array< T, Eigen::Dynamic, 1 > &sds, T &log_prob)
	This is the function that should be called prior to evaluating the density of any elliptical distribution. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	read_cov_matrix (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const Eigen::Array< T, Eigen::Dynamic, 1 > &sds, T &log_prob)
	A generally worse alternative to call prior to evaluating the density of an elliptical distribution. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	read_cov_matrix (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const Eigen::Array< T, Eigen::Dynamic, 1 > &sds)
	Builds a covariance matrix from CPCs and standard deviations. More...
template<typename T >
Eigen::Matrix< return_type_t< T >, Eigen::Dynamic, Eigen::Dynamic >	rep_matrix (const T &x, int m, int n)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	rep_matrix (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, int n)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	rep_matrix (const Eigen::Matrix< T, 1, Eigen::Dynamic > &rv, int m)
template<typename T >
Eigen::Matrix< return_type_t< T >, 1, Eigen::Dynamic >	rep_row_vector (const T &x, int m)
template<typename T >
Eigen::Matrix< return_type_t< T >, Eigen::Dynamic, 1 >	rep_vector (const T &x, int n)
template<typename T >
void	resize (T &x, std::vector< int > dims)
	Recursively resize the specified vector of vectors, which must bottom out at scalar values, Eigen vectors or Eigen matrices. More...
template<typename T >
apply_scalar_unary< round_fun, T >::return_t	round (const T &x)
	Vectorized version of round. More...
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic >	row (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t i)
	Return the specified row of the specified matrix, using start-at-1 indexing. More...
template<typename T , int R, int C>
int	rows (const Eigen::Matrix< T, R, C > &m)
	Return the number of rows in the specified matrix, vector, or row vector. More...
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< double, R1, 1 >	rows_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
	Returns the dot product of the specified vectors. More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, 1 >	rows_dot_self (const Eigen::Matrix< T, R, C > &x)
	Returns the dot product of each row of a matrix with itself. More...
template<int Cb>
Eigen::Matrix< double,-1, Cb >	scale_matrix_exp_multiply (const double &t, const Eigen::MatrixXd &A, const Eigen::Matrix< double,-1, Cb > &B)
	Return product of exp(At) and B, where A is a NxN double matrix, B is a NxCb double matrix, and t is a double. More...
template<typename Tt , typename Ta , typename Tb , int Cb>
Eigen::Matrix< stan::return_type_t< Tt, Ta, Tb >,-1, Cb >	scale_matrix_exp_multiply (const Tt &t, const Eigen::Matrix< Ta,-1,-1 > &A, const Eigen::Matrix< Tb,-1, Cb > &B)
	Return product of exp(At) and B, where A is a NxN matrix, B is a NxCb matrix and t is a scalar. More...
template<typename T >
return_type_t< T >	sd (const std::vector< T > &v)
	Returns the unbiased sample standard deviation of the coefficients in the specified column vector. More...
template<typename T , int R, int C>
return_type_t< T >	sd (const Eigen::Matrix< T, R, C > &m)
	Returns the unbiased sample standard deviation of the coefficients in the specified vector, row vector, or matrix. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	segment (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, size_t i, size_t n)
	Return the specified number of elements as a vector starting from the specified element - 1 of the specified vector. More...
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic >	segment (const Eigen::Matrix< T, 1, Eigen::Dynamic > &v, size_t i, size_t n)
template<typename T >
std::vector< T >	segment (const std::vector< T > &sv, size_t i, size_t n)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	simplex_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y)
	Return the simplex corresponding to the specified free vector. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	simplex_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y, T &lp)
	Return the simplex corresponding to the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	simplex_free (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
	Return an unconstrained vector that when transformed produces the specified simplex. More...
template<typename T >
apply_scalar_unary< sin_fun, T >::return_t	sin (const T &x)
	Vectorized version of sin(). More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	singular_values (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Return the vector of the singular values of the specified matrix in decreasing order of magnitude. More...
template<typename T >
apply_scalar_unary< sinh_fun, T >::return_t	sinh (const T &x)
	Vectorized version of sinh(). More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	softmax (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v)
	Return the softmax of the specified vector. More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	sort_asc (Eigen::Matrix< T, R, C > xs)
	Return the specified vector in ascending order. More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	sort_desc (Eigen::Matrix< T, R, C > xs)
	Return the specified vector in descending order. More...
template<bool ascending, typename C >
std::vector< int >	sort_indices (const C &xs)
	Return an integer array of indices of the specified container sorting the values in ascending or descending order based on the value of the first template prameter. More...
template<typename C >
std::vector< int >	sort_indices_asc (const C &xs)
	Return a sorted copy of the argument container in ascending order. More...
template<typename C >
std::vector< int >	sort_indices_desc (const C &xs)
	Return a sorted copy of the argument container in ascending order. More...
template<typename T >
apply_scalar_unary< sqrt_fun, T >::return_t	sqrt (const T &x)
	Vectorized version of sqrt(). More...
template<typename T >
apply_scalar_unary< square_fun, T >::return_t	square (const T &x)
	Vectorized version of square(). More...
template<int R, int C>
double	squared_distance (const Eigen::Matrix< double, R, C > &v1, const Eigen::Matrix< double, R, C > &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<int R1, int C1, int R2, int C2>
double	squared_distance (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
	Returns the squared distance between the specified vectors of the same dimensions. More...
template<typename T >
void	stan_print (std::ostream *o, const T &x)
template<typename T >
void	stan_print (std::ostream *o, const std::vector< T > &x)
template<typename T >
void	stan_print (std::ostream *o, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
template<typename T >
void	stan_print (std::ostream *o, const Eigen::Matrix< T, 1, Eigen::Dynamic > &x)
template<typename T >
void	stan_print (std::ostream *o, const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	sub_col (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t i, size_t j, size_t nrows)
	Return a nrows x 1 subcolumn starting at (i-1, j-1). More...
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic >	sub_row (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t i, size_t j, size_t ncols)
	Return a 1 x nrows subrow starting at (i-1, j-1). More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	subtract (const Eigen::Matrix< T1, R, C > &m1, const Eigen::Matrix< T2, R, C > &m2)
	Return the result of subtracting the second specified matrix from the first specified matrix. More...
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	subtract (const T1 &c, const Eigen::Matrix< T2, R, C > &m)
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< return_type_t< T1, T2 >, R, C >	subtract (const Eigen::Matrix< T1, R, C > &m, const T2 &c)
template<typename Derived >
Eigen::DenseBase< Derived >::Scalar	sum (const Eigen::DenseBase< Derived > &v)
	Returns the sum of the coefficients of the specified Eigen Matrix, Array or expression. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	tail (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, size_t n)
	Return the specified number of elements as a vector from the back of the specified vector. More...
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic >	tail (const Eigen::Matrix< T, 1, Eigen::Dynamic > &rv, size_t n)
	Return the specified number of elements as a row vector from the back of the specified row vector. More...
template<typename T >
std::vector< T >	tail (const std::vector< T > &sv, size_t n)
	Return the specified number of elements as a standard vector from the back of the specified standard vector. More...
template<typename T >
apply_scalar_unary< tan_fun, T >::return_t	tan (const T &x)
	Vectorized version of tan(). More...
template<typename T >
apply_scalar_unary< tanh_fun, T >::return_t	tanh (const T &x)
	Vectorized version of tanh(). More...
template<int R, int C>
Eigen::MatrixXd	tcrossprod (const Eigen::Matrix< double, R, C > &M)
	Returns the result of post-multiplying a matrix by its own transpose. More...
template<typename T >
apply_scalar_unary< tgamma_fun, T >::return_t	tgamma (const T &x)
	Vectorized version of tgamma(). More...
template<typename T , int R, int C>
std::vector< T >	to_array_1d (const Eigen::Matrix< T, R, C > &matrix)
template<typename T >
std::vector< T >	to_array_1d (const std::vector< T > &x)
template<typename T >
std::vector< typename scalar_type< T >::type >	to_array_1d (const std::vector< std::vector< T > > &x)
template<typename T >
std::vector< std::vector< T > >	to_array_2d (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &matrix)
template<typename T , int R, int C>
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const Eigen::Matrix< T, R, C > &x)
	Returns a matrix with dynamic dimensions constructed from an Eigen matrix which is either a row vector, column vector, or matrix. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< Eigen::Matrix< T, 1, Eigen::Dynamic > > &x)
	Returns a matrix representation of a standard vector of Eigen row vectors with the same dimensions and indexing order. More...
template<typename T >
Eigen::Matrix< return_type_t< T, double >, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< std::vector< T > > &x)
	Returns a matrix representation of the standard vector of standard vectors with the same dimensions and indexing order. More...
template<typename T , int R, int C>
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const Eigen::Matrix< T, R, C > &x, int m, int n)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< T > &x, int m, int n)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< int > &x, int m, int n)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
template<typename T , int R, int C>
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const Eigen::Matrix< T, R, C > &x, int m, int n, bool col_major)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
template<typename T >
Eigen::Matrix< return_type_t< T, double >, Eigen::Dynamic, Eigen::Dynamic >	to_matrix (const std::vector< T > &x, int m, int n, bool col_major)
	Returns a matrix representation of the vector in column-major order with the specified number of rows and columns. More...
template<typename T , int R, int C>
Eigen::Matrix< T, 1, Eigen::Dynamic >	to_row_vector (const Eigen::Matrix< T, R, C > &matrix)
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic >	to_row_vector (const std::vector< T > &vec)
Eigen::Matrix< double, 1, Eigen::Dynamic >	to_row_vector (const std::vector< int > &vec)
template<typename T , int R, int C>
Eigen::Matrix< T, Eigen::Dynamic, 1 >	to_vector (const Eigen::Matrix< T, R, C > &matrix)
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	to_vector (const std::vector< T > &vec)
Eigen::Matrix< double, Eigen::Dynamic, 1 >	to_vector (const std::vector< int > &vec)
template<typename T >
T	trace (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
	Returns the trace of the specified matrix. More...
template<typename T >
T	trace (const T &m)
template<typename T1 , typename T2 , typename T3 , int R1, int C1, int R2, int C2, int R3, int C3, typename = require_any_not_var_t<T1, T2, T3>>
return_type_t< T1, T2, T3 >	trace_gen_inv_quad_form_ldlt (const Eigen::Matrix< T1, R1, C1 > &D, const LDLT_factor< T2, R2, C2 > &A, const Eigen::Matrix< T3, R3, C3 > &B)
template<typename TD , int RD, int CD, typename TA , int RA, int CA, typename TB , int RB, int CB, typename = require_any_not_var_t<TD, TA, TB>>
return_type_t< TD, TA, TB >	trace_gen_quad_form (const Eigen::Matrix< TD, RD, CD > &D, const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)
	Return the trace of D times the quadratic form of B and A. More...
template<int RD, int CD, int RA, int CA, typename TB , int RB, int CB>
double	trace_gen_quad_form (const Eigen::Matrix< double, RD, CD > &D, const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< double, RB, CB > &B)
	Return the trace of D times the quadratic form of B and A. More...
template<typename T1 , typename T2 , int R2, int C2, int R3, int C3, typename = require_any_not_var_t<T1, T2>>
return_type_t< T1, T2 >	trace_inv_quad_form_ldlt (const LDLT_factor< T1, R2, C2 > &A, const Eigen::Matrix< T2, R3, C3 > &B)
template<int RA, int CA, int RB, int CB>
double	trace_quad_form (const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< double, RB, CB > &B)
	Compute trace(B^T A B). More...
template<typename T , int R, int C>
Eigen::Matrix< T, C, R >	transpose (const Eigen::Matrix< T, R, C > &m)
template<typename T >
apply_scalar_unary< trigamma_fun, T >::return_t	trigamma (const T &x)
	Return the elementwise application of trigamma() to specified argument container. More...
template<typename T >
apply_scalar_unary< trunc_fun, T >::return_t	trunc (const T &x)
	Return the elementwise application of trunc() to specified argument container. More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	unit_vector_constrain (const Eigen::Matrix< T, R, C > &y)
	Return the unit length vector corresponding to the free vector y. More...
template<typename T , int R, int C>
Eigen::Matrix< T, R, C >	unit_vector_constrain (const Eigen::Matrix< T, R, C > &y, T &lp)
	Return the unit length vector corresponding to the free vector y. More...
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 >	unit_vector_free (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
	Transformation of a unit length vector to a "free" vector However, we are just fixing the unidentified radius to 1. More...
template<typename T , int R, int C>
Eigen::Matrix< typename child_type< T >::type, R, C >	value_of (const Eigen::Matrix< T, R, C > &M)
	Convert a matrix of type T to a matrix of doubles. More...
template<int R, int C>
const Eigen::Matrix< double, R, C > &	value_of (const Eigen::Matrix< double, R, C > &x)
	Return the specified argument. More...
template<int R, int C>
const Eigen::Matrix< int, R, C > &	value_of (const Eigen::Matrix< int, R, C > &x)
	Return the specified argument. More...
template<typename T , int R, int C>
Eigen::Matrix< double, R, C >	value_of_rec (const Eigen::Matrix< T, R, C > &M)
	Convert a matrix of type T to a matrix of doubles. More...
template<int R, int C>
const Eigen::Matrix< double, R, C > &	value_of_rec (const Eigen::Matrix< double, R, C > &x)
	Return the specified argument. More...
template<typename T >
return_type_t< T >	variance (const std::vector< T > &v)
	Returns the sample variance (divide by length - 1) of the coefficients in the specified standard vector. More...
template<typename T , int R, int C>
return_type_t< T >	variance (const Eigen::Matrix< T, R, C > &m)
	Returns the sample variance (divide by length - 1) of the coefficients in the specified matrix. More...
template<typename F >
void	finite_diff_gradient (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::VectorXd &grad_fx, double epsilon=1e-03)
	Calculate the value and the gradient of the specified function at the specified argument using finite difference. More...
template<typename F >
void	finite_diff_gradient_auto (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::VectorXd &grad_fx)
	Calculate the value and the gradient of the specified function at the specified argument using finite difference. More...
template<typename F >
void	finite_diff_hessian (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::VectorXd &grad_fx, Eigen::MatrixXd &hess_fx, double epsilon=1e-03)
	Calculate the value and the Hessian of the specified function at the specified argument using second-order finite difference with the specified perturbation step size. More...
template<typename F >
void	finite_diff_hessian_auto (const F &f, const Eigen::VectorXd &x, double &fx, Eigen::VectorXd &grad_fx, Eigen::MatrixXd &hess_fx)
	Calculate the value and the Hessian of the specified function at the specified argument using second-order finite difference, automatically setting the stepsize between the function evaluations along a dimension. More...
template<typename F >
double	finite_diff_hessian_helper (const F &f, const Eigen::VectorXd &x, int i, double epsilon=1e-03)
	Return the subcalculation required by finite_diff_hessian and finite_diff_hessian_auto. More...
template<int call_id, typename F , typename T_shared_param , typename T_job_param >
Eigen::Matrix< return_type_t< T_shared_param, T_job_param >, Eigen::Dynamic, 1 >	map_rect (const Eigen::Matrix< T_shared_param, Eigen::Dynamic, 1 > &shared_params, const std::vector< Eigen::Matrix< T_job_param, Eigen::Dynamic, 1 >> &job_params, const std::vector< std::vector< double >> &x_r, const std::vector< std::vector< int >> &x_i, std::ostream *msgs=nullptr)
	Map N function evaluations to parameters and data which are in rectangular format. More...
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	bernoulli_logit_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	bernoulli_logit_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<bool propto, typename T_y , typename T_x_scalar , int T_x_rows, typename T_alpha , typename T_beta >
return_type_t< T_x_scalar, T_alpha, T_beta >	bernoulli_logit_glm_lpmf (const T_y &y, const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &x, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Bernoulli distribution and logit link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_beta, T_alpha >	bernoulli_logit_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<typename T_x , typename T_alpha , typename T_beta , class RNG >
VectorBuilder< true, int, T_alpha >::type	bernoulli_logit_glm_rng (const T_x &x, const T_alpha &alpha, const T_beta &beta, RNG &rng)
	Returns a draw from the Generalized Linear Model (GLM) with Bernoulli distribution and logit link function. More...
template<bool propto, typename T_prob >
return_type_t< T_prob >	categorical_log (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<typename T_prob >
return_type_t< T_prob >	categorical_log (const typename math::index_type< Eigen::VectorXd >::type n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<bool propto, typename T_prob >
return_type_t< T_prob >	categorical_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<typename T_prob >
return_type_t< T_prob >	categorical_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<bool propto, typename T_y , typename T_x_scalar , int T_x_rows, typename T_alpha_scalar , typename T_beta_scalar >
return_type_t< T_x_scalar, T_alpha_scalar, T_beta_scalar >	categorical_logit_glm_lpmf (const T_y &y, const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &x, const Eigen::Matrix< T_alpha_scalar, Eigen::Dynamic, 1 > &alpha, const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, Eigen::Dynamic > &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with categorical distribution and logit (softmax) link function. More...
template<typename T_y , typename T_x_scalar , int T_x_rows, typename T_alpha_scalar , typename T_beta_scalar >
return_type_t< T_x_scalar, T_alpha_scalar, T_beta_scalar >	categorical_logit_glm_lpmf (const T_y &y, const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &x, const Eigen::Matrix< T_alpha_scalar, Eigen::Dynamic, 1 > &alpha, const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, Eigen::Dynamic > &beta)
template<bool propto, typename T_prob >
return_type_t< T_prob >	categorical_logit_log (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
template<typename T_prob >
return_type_t< T_prob >	categorical_logit_log (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
template<bool propto, typename T_prob >
return_type_t< T_prob >	categorical_logit_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
template<typename T_prob >
return_type_t< T_prob >	categorical_logit_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
template<bool propto, typename T_prob >
return_type_t< T_prob >	categorical_logit_lpmf (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
template<typename T_prob >
return_type_t< T_prob >	categorical_logit_lpmf (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
template<bool propto, typename T_prob >
return_type_t< T_prob >	categorical_logit_lpmf (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
template<typename T_prob >
return_type_t< T_prob >	categorical_logit_lpmf (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
template<class RNG >
int	categorical_logit_rng (const Eigen::VectorXd &beta, RNG &rng)
	Return a draw from a Categorical distribution given a a vector of unnormalized log probabilities and a psuedo-random number generator. More...
template<bool propto, typename T_prob >
return_type_t< T_prob >	categorical_lpmf (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<typename T_prob >
return_type_t< T_prob >	categorical_lpmf (const typename math::index_type< Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > >::type n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<bool propto, typename T_prob >
return_type_t< T_prob >	categorical_lpmf (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<typename T_prob >
return_type_t< T_prob >	categorical_lpmf (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<class RNG >
int	categorical_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &theta, RNG &rng)
template<bool propto, typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_log (const T_prob &theta, const T_prior_size &alpha)
	The log of the Dirichlet density for the given theta and a vector of prior sample sizes, alpha. More...
template<typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_log (const T_prob &theta, const T_prior_size &alpha)
template<bool propto, typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_lpdf (const T_prob &theta, const T_prior_size &alpha)
	The log of the Dirichlet density for the given theta and a vector of prior sample sizes, alpha. More...
template<typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_lpdf (const T_prob &theta, const T_prior_size &alpha)
template<bool propto, typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_lpmf (const T_prob &theta, const T_prior_size &alpha)
template<typename T_prob , typename T_prior_size >
return_type_t< T_prob, T_prior_size >	dirichlet_lpmf (const T_prob &theta, const T_prior_size &alpha)
template<class RNG >
Eigen::VectorXd	dirichlet_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &alpha, RNG &rng)
	Return a draw from a Dirichlet distribution with specified parameters and pseudo-random number generator. More...
template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, return_type_t< T_F, T_G, T_V, T_W, T_m0, T_C0 > >	gaussian_dlm_obs_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
	The log of a Gaussian dynamic linear model (GDLM). More...
template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, return_type_t< T_F, T_G, T_V, T_W, T_m0, T_C0 > >	gaussian_dlm_obs_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, return_type_t< T_F, T_G, T_V, T_W, T_m0, T_C0 > >	gaussian_dlm_obs_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
	The log of a Gaussian dynamic linear model (GDLM) with uncorrelated observation disturbances. More...
template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, return_type_t< T_F, T_G, T_V, T_W, T_m0, T_C0 > >	gaussian_dlm_obs_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, return_type_t< T_F, T_G, T_V, T_W, T_m0, T_C0 > >	gaussian_dlm_obs_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
	The log of a Gaussian dynamic linear model (GDLM). More...
template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, return_type_t< T_F, T_G, T_V, T_W, T_m0, T_C0 > >	gaussian_dlm_obs_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, return_type_t< T_F, T_G, T_V, T_W, T_m0, T_C0 > >	gaussian_dlm_obs_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
	The log of a Gaussian dynamic linear model (GDLM) with uncorrelated observation disturbances. More...
template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type_t< T_y, return_type_t< T_F, T_G, T_V, T_W, T_m0, T_C0 > >	gaussian_dlm_obs_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
template<class RNG >
Eigen::MatrixXd	gaussian_dlm_obs_rng (const Eigen::MatrixXd &F, const Eigen::MatrixXd &G, const Eigen::MatrixXd &V, const Eigen::MatrixXd &W, const Eigen::VectorXd &m0, const Eigen::MatrixXd &C0, const int T, RNG &rng)
	Simulate random draw from Gaussian dynamic linear model (GDLM). More...
template<class RNG >
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >	gaussian_dlm_obs_rng (const Eigen::MatrixXd &F, const Eigen::MatrixXd &G, const Eigen::VectorXd &V, const Eigen::MatrixXd &W, const Eigen::VectorXd &m0, const Eigen::MatrixXd &C0, const int T, RNG &rng)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	inv_wishart_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
	The log of the Inverse-Wishart density for the given W, degrees of freedom, and scale matrix. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	inv_wishart_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	inv_wishart_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
	The log of the Inverse-Wishart density for the given W, degrees of freedom, and scale matrix. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	inv_wishart_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
template<class RNG >
Eigen::MatrixXd	inv_wishart_rng (double nu, const Eigen::MatrixXd &S, RNG &rng)
template<bool propto, typename T_covar , typename T_shape >
return_type_t< T_covar, T_shape >	lkj_corr_cholesky_log (const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const T_shape &eta)
template<typename T_covar , typename T_shape >
return_type_t< T_covar, T_shape >	lkj_corr_cholesky_log (const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const T_shape &eta)
template<bool propto, typename T_covar , typename T_shape >
return_type_t< T_covar, T_shape >	lkj_corr_cholesky_lpdf (const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const T_shape &eta)
template<typename T_covar , typename T_shape >
return_type_t< T_covar, T_shape >	lkj_corr_cholesky_lpdf (const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const T_shape &eta)
template<class RNG >
Eigen::MatrixXd	lkj_corr_cholesky_rng (size_t K, double eta, RNG &rng)
template<bool propto, typename T_y , typename T_shape >
return_type_t< T_y, T_shape >	lkj_corr_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_shape &eta)
template<typename T_y , typename T_shape >
return_type_t< T_y, T_shape >	lkj_corr_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_shape &eta)
template<typename T_shape >
return_type_t< double, T_shape >	do_lkj_constant (const T_shape &eta, const unsigned int &K)
template<bool propto, typename T_y , typename T_shape >
return_type_t< T_y, T_shape >	lkj_corr_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_shape &eta)
template<typename T_y , typename T_shape >
return_type_t< T_y, T_shape >	lkj_corr_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_shape &eta)
template<class RNG >
Eigen::MatrixXd	lkj_corr_rng (size_t K, double eta, RNG &rng)
	Return a random correlation matrix (symmetric, positive definite, unit diagonal) of the specified dimensionality drawn from the LKJ distribution with the specified degrees of freedom using the specified random number generator. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &sigma, const T_shape &eta)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &sigma, const T_shape &eta)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &sigma, const T_shape &eta)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &sigma, const T_shape &eta)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	lkj_cov_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
template<bool propto, typename T_y , typename T_Mu , typename T_Sigma , typename T_D >
return_type_t< T_y, T_Mu, T_Sigma, T_D >	matrix_normal_prec_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &Mu, const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &D)
	The log of the matrix normal density for the given y, mu, Sigma and D where Sigma and D are given as precision matrices, not covariance matrices. More...
template<typename T_y , typename T_Mu , typename T_Sigma , typename T_D >
return_type_t< T_y, T_Mu, T_Sigma, T_D >	matrix_normal_prec_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &Mu, const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &D)
template<bool propto, typename T_y , typename T_Mu , typename T_Sigma , typename T_D >
return_type_t< T_y, T_Mu, T_Sigma, T_D >	matrix_normal_prec_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &Mu, const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &D)
	The log of the matrix normal density for the given y, mu, Sigma and D where Sigma and D are given as precision matrices, not covariance matrices. More...
template<typename T_y , typename T_Mu , typename T_Sigma , typename T_D >
return_type_t< T_y, T_Mu, T_Sigma, T_D >	matrix_normal_prec_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &Mu, const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &D)
template<class RNG >
Eigen::MatrixXd	matrix_normal_prec_rng (const Eigen::MatrixXd &Mu, const Eigen::MatrixXd &Sigma, const Eigen::MatrixXd &D, RNG &rng)
	Sample from the the matrix normal distribution for the given Mu, Sigma and D where Sigma and D are given as precision matrices, not covariance matrices. More...
template<bool propto, typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_cholesky_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
	The log of a multivariate Gaussian Process for the given y, w, and a Cholesky factor L of the kernel matrix Sigma. More...
template<typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_cholesky_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
template<bool propto, typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_cholesky_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
	The log of a multivariate Gaussian Process for the given y, w, and a Cholesky factor L of the kernel matrix Sigma. More...
template<typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_cholesky_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
template<bool propto, typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
	The log of a multivariate Gaussian Process for the given y, Sigma, and w. More...
template<typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
template<bool propto, typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
	The log of a multivariate Gaussian Process for the given y, Sigma, and w. More...
template<typename T_y , typename T_covar , typename T_w >
return_type_t< T_y, T_covar, T_w >	multi_gp_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_cholesky_log (const T_y &y, const T_loc &mu, const T_covar &L)
	The log of the multivariate normal density for the given y, mu, and a Cholesky factor L of the variance matrix. More...
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_cholesky_log (const T_y &y, const T_loc &mu, const T_covar &L)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_cholesky_lpdf (const T_y &y, const T_loc &mu, const T_covar &L)
	The log of the multivariate normal density for the given y, mu, and a Cholesky factor L of the variance matrix. More...
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_cholesky_lpdf (const T_y &y, const T_loc &mu, const T_covar &L)
template<typename T_loc , class RNG >
StdVectorBuilder< true, Eigen::VectorXd, T_loc >::type	multi_normal_cholesky_rng (const T_loc &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &L, RNG &rng)
	Return a multivariate normal random variate with the given location and Cholesky factorization of the covariance using the specified random number generator. More...
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_lpdf (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_lpdf (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_prec_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_prec_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_prec_lpdf (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_y , typename T_loc , typename T_covar >
return_type_t< T_y, T_loc, T_covar >	multi_normal_prec_lpdf (const T_y &y, const T_loc &mu, const T_covar &Sigma)
template<typename T_loc , class RNG >
StdVectorBuilder< true, Eigen::VectorXd, T_loc >::type	multi_normal_prec_rng (const T_loc &mu, const Eigen::MatrixXd &S, RNG &rng)
	Return a multivariate normal random variate with the given location and precision using the specified random number generator. More...
template<typename T_loc , class RNG >
StdVectorBuilder< true, Eigen::VectorXd, T_loc >::type	multi_normal_rng (const T_loc &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &S, RNG &rng)
	Return a multivariate normal random variate with the given location and covariance using the specified random number generator. More...
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	multi_student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
	Return the log of the multivariate Student t distribution at the specified arguments. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	multi_student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	multi_student_t_lpdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
	Return the log of the multivariate Student t distribution at the specified arguments. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	multi_student_t_lpdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
template<typename T_loc , class RNG >
StdVectorBuilder< true, Eigen::VectorXd, T_loc >::type	multi_student_t_rng (double nu, const T_loc &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &S, RNG &rng)
	Return a multivariate student-t random variate with the given degrees of freedom location and covariance using the specified random number generator. More...
template<bool propto, typename T_prob >
return_type_t< T_prob >	multinomial_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<typename T_prob >
return_type_t< T_prob >	multinomial_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<bool propto, typename T_prob >
return_type_t< T_prob >	multinomial_lpmf (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<typename T_prob >
return_type_t< T_prob >	multinomial_lpmf (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
template<class RNG >
std::vector< int >	multinomial_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &theta, int N, RNG &rng)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_x, T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_x, T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
template<bool propto, typename T_y , typename T_x_scalar , int T_x_rows, typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_x_scalar, T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_lpmf (const T_y &y, const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
	Returns the log PMF of the Generalized Linear Model (GLM) with Negative-Binomial-2 distribution and log link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision >
return_type_t< T_x, T_alpha, T_beta, T_precision >	neg_binomial_2_log_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_precision &phi)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_y, T_x, T_alpha, T_beta, T_scale >	normal_id_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_y, T_x, T_alpha, T_beta, T_scale >	normal_id_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
template<bool propto, typename T_y , typename T_x_scalar , int T_x_rows, typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_y, T_x_scalar, T_alpha, T_beta, T_scale >	normal_id_glm_lpdf (const T_y &y, const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
	Returns the log PDF of the Generalized Linear Model (GLM) with Normal distribution and id link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale >
return_type_t< T_y, T_x, T_alpha, T_beta, T_scale >	normal_id_glm_lpdf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta, const T_scale &sigma)
template<bool propto, typename T_y , typename T_x_scalar , int T_x_rows, typename T_beta_scalar , typename T_cuts_scalar >
stan::return_type_t< T_x_scalar, T_beta_scalar, T_cuts_scalar >	ordered_logistic_glm_lpmf (const T_y &y, const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &x, const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, 1 > &beta, const Eigen::Matrix< T_cuts_scalar, Eigen::Dynamic, 1 > &cuts)
	Returns the log PMF of the ordinal regression Generalized Linear Model (GLM). More...
template<typename T_y , typename T_x_scalar , int T_x_rows, typename T_beta_scalar , typename T_cuts_scalar >
stan::return_type_t< T_x_scalar, T_beta_scalar, T_cuts_scalar >	ordered_logistic_glm_lpmf (const T_y &y, const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &x, const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, 1 > &beta, const Eigen::Matrix< T_cuts_scalar, Eigen::Dynamic, 1 > &cuts)
template<bool propto, typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_logistic_log (const T_y &y, const T_loc &lambda, const T_cut &c)
	Returns the (natural) log probability of the integer/s given the vector of continuous location/s and specified cutpoints in an ordered logistic model. More...
template<typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_logistic_log (const T_y &y, const T_loc &lambda, const T_cut &c)
template<bool propto, typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_logistic_lpmf (const T_y &y, const T_loc &lambda, const T_cut &c)
	Returns the (natural) log probability of the specified array of integers given the vector of continuous locations and specified cutpoints in an ordered logistic model. More...
template<typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_logistic_lpmf (const T_y &y, const T_loc &lambda, const T_cut &c)
template<class RNG >
int	ordered_logistic_rng (double eta, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &c, RNG &rng)
template<bool propto, typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_log (const T_y &y, const T_loc &lambda, const T_cut &c)
	Returns the (natural) log probability of the integer/s given the vector of continuous location/s and specified cutpoints in an ordered probit model. More...
template<typename T_y , typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_log (const T_y &y, const T_loc &lambda, const T_cut &c)
template<bool propto, typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_lpmf (int y, const T_loc &lambda, const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &c)
	Returns the (natural) log probability of the specified integer outcome given the continuous location and specified cutpoints in an ordered probit model. More...
template<typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_lpmf (int y, const T_loc &lambda, const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &c)
template<bool propto, typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_lpmf (const std::vector< int > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &lambda, const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &c)
	Returns the (natural) log probability of the specified array of integers given the vector of continuous locations and specified cutpoints in an ordered probit model. More...
template<typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_lpmf (const std::vector< int > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &lambda, const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &c)
template<bool propto, typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_lpmf (const std::vector< int > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &lambda, const std::vector< Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > > &c)
	Returns the (natural) log probability of the specified array of integers given the vector of continuous locations and array of specified cutpoints in an ordered probit model. More...
template<typename T_loc , typename T_cut >
return_type_t< T_loc, T_cut >	ordered_probit_lpmf (const std::vector< int > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &lambda, const std::vector< Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > > &c)
template<class RNG >
int	ordered_probit_rng (double eta, const Eigen::VectorXd &c, RNG &rng)
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	poisson_log_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	poisson_log_glm_log (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<bool propto, typename T_y , typename T_x_scalar , int T_x_rows, typename T_alpha , typename T_beta >
return_type_t< T_x_scalar, T_alpha, T_beta >	poisson_log_glm_lpmf (const T_y &y, const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &x, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Poisson distribution and log link function. More...
template<typename T_y , typename T_x , typename T_alpha , typename T_beta >
return_type_t< T_x, T_alpha, T_beta >	poisson_log_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	wishart_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
	The log of the Wishart density for the given W, degrees of freedom, and scale matrix. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	wishart_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	wishart_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
	The log of the Wishart density for the given W, degrees of freedom, and scale matrix. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	wishart_lpdf (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
template<class RNG >
Eigen::MatrixXd	wishart_rng (double nu, const Eigen::MatrixXd &S, RNG &rng)
template<typename T , typename = require_stan_scalar_t<T>>
const T &	as_array_or_scalar (const T &v)
	Returns specified input value. More...
template<typename T , typename = require_eigen_t<T>>
auto	as_array_or_scalar (const T &v)
	Converts a matrix type to an array. More...
template<typename T >
Eigen::Map< const Eigen::Array< T, Eigen::Dynamic, 1 > >	as_array_or_scalar (const std::vector< T > &v)
	Converts a std::vector type to an array. More...
template<typename T >
const T &	as_column_vector_or_scalar (const T &a)
	Converts input argument to a column vector or a scalar. More...
template<typename T >
const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	as_column_vector_or_scalar (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &a)
	Converts input argument to a column vector or a scalar. More...
template<typename T >
Eigen::Map< const Eigen::Matrix< T, Eigen::Dynamic, 1 > >	as_column_vector_or_scalar (const Eigen::Matrix< T, 1, Eigen::Dynamic > &a)
	Converts input argument to a column vector or a scalar. More...
template<typename T >
Eigen::Map< const Eigen::Matrix< T, Eigen::Dynamic, 1 > >	as_column_vector_or_scalar (const std::vector< T > &a)
	Converts input argument to a column vector or a scalar. More...
template<typename T_desired , typename T_actual , typename = std::enable_if_t<std::is_convertible<T_actual, T_desired>::value && !is_eigen<T_desired>::value>>
T_actual &&	forward_as (T_actual &&a)
	Assume which type we get. More...
template<typename T_desired , typename T_actual , typename = std::enable_if_t<!std::is_convertible<T_actual, T_desired>::value>>
T_desired	forward_as (const T_actual &a)
	Assume which type we get. More...
template<typename T_desired , typename T_actual , typename = std::enable_if_t< std::is_convertible<T_actual, T_desired>::value && static_cast< int>(T_desired::RowsAtCompileTime) == static_cast<int>(T_actual::RowsAtCompileTime) && static_cast<int>(T_desired::ColsAtCompileTime) == static_cast<int>(T_actual::ColsAtCompileTime)>, typename = void>
T_actual &&	forward_as (T_actual &&a)
	Assume which type we get. More...
template<typename T_desired , typename T_actual , typename = std::enable_if_t< !std::is_convertible<T_actual, T_desired>::value || static_cast<int>(T_desired::RowsAtCompileTime) != static_cast<int>(T_actual::RowsAtCompileTime) || static_cast<int>(T_desired::ColsAtCompileTime) != static_cast<int>(T_actual::ColsAtCompileTime)>, typename = void>
T_desired	forward_as (const T_actual &a)
	Assume which type we get. More...
template<std::size_t N, class F >
constexpr auto	index_apply (F f)
	Calls given callable with an index sequence. More...
template<typename T1 , typename T2 >
size_t	max_size (const T1 &x1, const T2 &x2)
template<typename T1 , typename T2 , typename T3 >
size_t	max_size (const T1 &x1, const T2 &x2, const T3 &x3)
template<typename T1 , typename T2 , typename T3 , typename T4 >
size_t	max_size (const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4)
template<typename T1 , typename T2 , typename T3 , typename T4 , typename T5 >
size_t	max_size (const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4, const T5 &x5)
template<typename T1 , typename T2 >
size_t	max_size_mvt (const T1 &x1, const T2 &x2)
template<typename T1 , typename T2 , typename T3 >
size_t	max_size_mvt (const T1 &x1, const T2 &x2, const T3 &x3)
template<typename T1 , typename T2 , typename T3 , typename T4 >
size_t	max_size_mvt (const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4)
template<typename T , typename = require_stan_scalar_t<T>>
size_t	size (const T &)
	Returns the length of primitive scalar types that are always of length 1. More...
template<typename T , typename = require_not_stan_scalar_t<T>, typename = void>
size_t	size (const T &m)
	Returns the size of the provided Eigen matrix, expression or std::vector. More...
template<typename T >
size_t	size_mvt (const T &)
	Provides the size of a multivariate argument. More...
template<typename T , int R, int C>
size_t	size_mvt (const Eigen::Matrix< T, R, C > &)
template<typename T , int R, int C>
size_t	size_mvt (const std::vector< Eigen::Matrix< T, R, C > > &x)
template<typename T_a1 , typename T_a2 , typename T_b1 , typename T_z >
void	check_2F1_converges (const char *function, const T_a1 &a1, const T_a2 &a2, const T_b1 &b1, const T_z &z)
	Check if the hypergeometric function (2F1) called with supplied arguments will converge, assuming arguments are finite values. More...
template<typename T_a1 , typename T_a2 , typename T_a3 , typename T_b1 , typename T_b2 , typename T_z >
void	check_3F2_converges (const char *function, const T_a1 &a1, const T_a2 &a2, const T_a3 &a3, const T_b1 &b1, const T_b2 &b2, const T_z &z)
	Check if the hypergeometric function (3F2) called with supplied arguments will converge, assuming arguments are finite values. More...
template<typename T_y , typename T_low , typename T_high >
void	check_bounded (const char *function, const char *name, const T_y &y, const T_low &low, const T_high &high)
	Check if the value is between the low and high values, inclusively. More...
template<typename T >
void	check_consistent_size (const char *function, const char *name, const T &x, size_t expected_size)
	Check if the dimension of x is consistent, which is defined to be expected_size if x is a vector or 1 if x is not a vector. More...
template<typename T1 , typename T2 >
void	check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2)
	Check if the dimension of x1 is consistent with x2. More...
template<typename T1 , typename T2 , typename T3 >
void	check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2, const char *name3, const T3 &x3)
	Check if the dimension of x1, x2, and x3 areconsistent. More...
template<typename T1 , typename T2 , typename T3 , typename T4 >
void	check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2, const char *name3, const T3 &x3, const char *name4, const T4 &x4)
	Check if the dimension of x1, x2, x3, and x4are consistent. More...
template<typename T1 , typename T2 , typename T3 , typename T4 , typename T5 >
void	check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2, const char *name3, const T3 &x3, const char *name4, const T4 &x4, const char *name5, const T5 &x5)
template<typename T_y >
void	check_finite (const char *function, const char *name, const T_y &y)
	Check if y is finite. More...
template<typename T_y , typename T_low >
void	check_greater (const char *function, const char *name, const T_y &y, const T_low &low)
	Check if y is strictly greater than low. More...
template<typename T_y , typename T_low >
void	check_greater_or_equal (const char *function, const char *name, const T_y &y, const T_low &low)
	Check if y is greater or equal than low. More...
template<typename T_y , typename T_high >
void	check_less (const char *function, const char *name, const T_y &y, const T_high &high)
	Check if y is strictly less than high. More...
template<typename T_y , typename T_high >
void	check_less_or_equal (const char *function, const char *name, const T_y &y, const T_high &high)
	Check if y is less or equal to high. More...
template<typename T_y >
void	check_nonnegative (const char *function, const char *name, const T_y &y)
	Check if y is non-negative. More...
template<typename T_y >
void	check_not_nan (const char *function, const char *name, const T_y &y)
	Check if y is not NaN. More...
template<typename T_y >
void	check_positive (const char *function, const char *name, const T_y &y)
	Check if y is positive. More...
void	check_positive (const char *function, const char *name, const char *expr, int size)
	Check if size is positive. More...
template<typename T_y >
void	check_positive_finite (const char *function, const char *name, const T_y &y)
	Check if y is positive and finite. More...
template<typename T_size1 , typename T_size2 >
void	check_size_match (const char *function, const char *name_i, T_size1 i, const char *name_j, T_size2 j)
	Check if the provided sizes match. More...
template<typename T_size1 , typename T_size2 >
void	check_size_match (const char *function, const char *expr_i, const char *name_i, T_size1 i, const char *expr_j, const char *name_j, T_size2 j)
	Check if the provided sizes match. More...
template<typename T >
void	domain_error (const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
template<typename T >
void	domain_error (const char *function, const char *name, const T &y, const char *msg1)
template<typename T >
void	domain_error_vec (const char *function, const char *name, const T &y, size_t i, const char *msg1, const char *msg2)
template<typename T >
void	domain_error_vec (const char *function, const char *name, const T &y, size_t i, const char *msg1)
template<typename T >
void	invalid_argument (const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
	Throw an invalid_argument exception with a consistently formatted message. More...
template<typename T >
void	invalid_argument (const char *function, const char *name, const T &y, const char *msg1)
	Throw an invalid_argument exception with a consistently formatted message. More...
template<typename T >
void	invalid_argument_vec (const char *function, const char *name, const T &y, size_t i, const char *msg1, const char *msg2)
	Throw an invalid argument exception with a consistently formatted message. More...
template<typename T >
void	invalid_argument_vec (const char *function, const char *name, const T &y, size_t i, const char *msg)
	Throw an invalid argument exception with a consistently formatted message. More...
template<typename T_y , typename T_high >
bool	is_less_or_equal (const T_y &y, const T_high &high)
	Return true if y is less or equal to high. More...
template<typename T_y >
bool	is_not_nan (const T_y &y)
	Return true if y is not NaN. More...
template<typename T_y >
bool	is_positive (const T_y &y)
	Return true if y is positive. More...
bool	is_positive (int size)
	Return true if size is positive. More...
template<typename T_y >
bool	is_scal_finite (const T_y &y)
	Return true if y is finite. More...
template<typename T_size1 , typename T_size2 >
bool	is_size_match (T_size1 i, T_size2 j)
	Return true if the provided sizes match. More...
void	out_of_range (const char *function, int max, int index, const char *msg1="", const char *msg2="")
	Throw an out_of_range exception with a consistently formatted message. More...
void	system_error (const char *function, const char *name, const int &y, const char *msg1, const char *msg2)
	Throw a system error with a consistently formatted message. More...
void	system_error (const char *function, const char *name, const int &y, const char *msg1)
	Throw a system error with a consistently formatted message. More...
template<typename T >
void	throw_domain_error (const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
	Throw a domain error with a consistently formatted message. More...
template<typename T >
void	throw_domain_error (const char *function, const char *name, const T &y, const char *msg1)
	Throw a domain error with a consistently formatted message. More...
template<typename T >
void	throw_domain_error_vec (const char *function, const char *name, const T &y, size_t i, const char *msg1, const char *msg2)
	Throw a domain error with a consistently formatted message. More...
template<typename T >
void	throw_domain_error_vec (const char *function, const char *name, const T &y, size_t i, const char *msg)
	Throw a domain error with a consistently formatted message. More...
double	abs (double x)
	Return floating-point absolute value. More...
double	acosh (double x)
	Return the inverse hyperbolic cosine of the specified value. More...
double	acosh (int x)
	Integer version of acosh. More...
template<typename T >
bool	as_bool (const T &x)
	Return true if the argument is not equal to zero (in the != operator sense) and false otherwise. More...
double	asinh (double x)
	Return the inverse hyperbolic sine of the specified value. More...
double	asinh (int x)
	Integer version of asinh. More...
double	atanh (double x)
	Return the inverse hyperbolic tangent of the specified value. More...
double	atanh (int x)
	Integer version of atanh. More...
template<typename T2 >
T2	bessel_first_kind (int v, const T2 z)
	\[ \mbox{bessel\_first\_kind}(v, x) = \begin{cases} J_v(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \] More...
template<typename T2 >
T2	bessel_second_kind (int v, const T2 z)
	\[ \mbox{bessel\_second\_kind}(v, x) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ Y_v(x) & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \] More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	beta (const T1 a, const T2 b)
	Return the beta function applied to the specified arguments. More...
template<typename T >
T	binary_log_loss (int y, const T &y_hat)
	Returns the log loss function for binary classification with specified reference and response values. More...
template<typename T_N , typename T_n >
return_type_t< T_N, T_n >	binomial_coefficient_log (const T_N N, const T_n n)
	Return the log of the binomial coefficient for the specified arguments. More...
double	cbrt (double x)
	Return the cube root of the specified value. More...
double	cbrt (int x)
	Integer version of cbrt. More...
int	choose (int n, int k)
	Return the binomial coefficient for the specified integer arguments. More...
double	pi ()
	Return the value of pi. More...
double	e ()
	Return the base of the natural logarithm. More...
double	sqrt2 ()
	Return the square root of two. More...
double	log10 ()
	Return natural logarithm of ten. More...
double	positive_infinity ()
	Return positive infinity. More...
double	negative_infinity ()
	Return negative infinity. More...
double	not_a_number ()
	Return (quiet) not-a-number. More...
double	machine_precision ()
	Returns the difference between 1.0 and the next value representable. More...
template<typename T >
T	corr_constrain (const T &x)
	Return the result of transforming the specified scalar to have a valid correlation value between -1 and 1 (inclusive). More...
template<typename T >
T	corr_constrain (const T &x, T &lp)
	Return the result of transforming the specified scalar to have a valid correlation value between -1 and 1 (inclusive). More...
template<typename T >
T	corr_free (const T &y)
	Return the unconstrained scalar that when transformed to a valid correlation produces the specified value. More...
double	digamma (double x)
	Return the derivative of the log gamma function at the specified value. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	distance (const T1 &x1, const T2 &x2)
	Returns the distance between two scalars. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	divide (const T1 &x, const T2 &y)
	Return the division of the first scalar by the second scalar. More...
int	divide (int x, int y)
double	erf (double x)
	Return the error function of the specified value. More...
double	erf (int x)
	Return the error function of the specified argument. More...
double	erfc (double x)
	Return the complementary error function of the specified value. More...
double	erfc (int x)
	Return the error function of the specified argument. More...
double	exp (int x)
	Return the natural exponential of the specified argument. More...
template<typename T , typename = require_arithmetic_t<T>>
double	exp2 (T y)
	Return the exponent base 2 of the specified argument (C99, C++11). More...
double	expm1 (double x)
	Return the natural exponentiation of x minus one. More...
double	expm1 (int x)
	Integer version of expm1. More...
template<typename T >
T	F32 (const T &a1, const T &a2, const T &a3, const T &b1, const T &b2, const T &z, double precision=1e-6, int max_steps=1e5)
	Hypergeometric function (3F2). More...
template<typename T >
return_type_t< T >	falling_factorial (const T &x, int n)
	Return the falling factorial function evaluated at the inputs. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
double	fdim (T1 x, T2 y)
	Return the positive difference of the specified values (C++11). More...
template<typename T , typename S >
void	fill (T &x, const S &y)
	Fill the specified container with the specified value. More...
double	finite_diff_stepsize (double u)
	Return the stepsize for finite difference evaluations at the specified scalar. More...
template<typename T1 , typename T2 , typename T3 , typename = require_all_arithmetic_t<T1, T2, T3>>
double	fma (T1 x, T2 y, T3 z)
	Return the product of the first two arguments plus the third argument. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
double	fmax (T1 x, T2 y)
	Return the greater of the two specified arguments. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
double	fmin (T1 x, T2 y)
	Return the lesser of the two specified arguments. More...
double	gamma_p (double z, double a)
	Return the value of the normalized, lower-incomplete gamma function applied to the specified argument. More...
double	gamma_q (double x, double a)
	\[ \mbox{gamma\_q}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ Q(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \] More...
template<typename T >
void	grad_2F1 (T &g_a1, T &g_b1, const T &a1, const T &a2, const T &b1, const T &z, const T &precision=1e-10, int max_steps=1e5)
	Gradients of the hypergeometric function, 2F1. More...
template<typename T >
void	grad_F32 (T *g, const T &a1, const T &a2, const T &a3, const T &b1, const T &b2, const T &z, const T &precision=1e-6, int max_steps=1e5)
	Gradients of the hypergeometric function, 3F2. More...
void	grad_inc_beta (double &g1, double &g2, double a, double b, double z)
template<typename T >
void	grad_reg_inc_beta (T &g1, T &g2, const T &a, const T &b, const T &z, const T &digammaA, const T &digammaB, const T &digammaSum, const T &betaAB)
	Computes the gradients of the regularized incomplete beta function. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	grad_reg_inc_gamma (T1 a, T2 z, T1 g, T1 dig, double precision=1e-6, int max_steps=1e5)
	Gradient of the regularized incomplete gamma functions igamma(a, z) More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	grad_reg_lower_inc_gamma (const T1 &a, const T2 &z, double precision=1e-10, int max_steps=1e5)
	Computes the gradient of the lower regularized incomplete gamma function. More...
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
double	hypot (T1 x, T2 y)
	Return the length of the hypoteneuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11). More...
double	ibeta (double a, double b, double x)
	The normalized incomplete beta function of a, b, and x. More...
template<typename T >
T	identity_constrain (const T &x)
	Returns the result of applying the identity constraint transform to the input. More...
template<typename T >
T	identity_constrain (const T &x, T &)
	Returns the result of applying the identity constraint transform to the input and increments the log probability reference with the log absolute Jacobian determinant. More...
template<typename T >
T	identity_free (const T &y)
	Returns the result of applying the inverse of the identity constraint transform to the input. More...
template<typename T_true , typename T_false >
return_type_t< T_true, T_false >	if_else (const bool c, const T_true y_true, const T_false y_false)
	Return the second argument if the first argument is true and otherwise return the second argument. More...
double	inc_beta (double a, double b, double x)
	The normalized incomplete beta function of a, b, with outcome x. More...
template<typename T >
T	inc_beta_dda (T a, T b, T z, T digamma_a, T digamma_ab)
	Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to a. More...
template<typename T >
T	inc_beta_ddb (T a, T b, T z, T digamma_b, T digamma_ab)
	Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to b. More...
template<typename T >
T	inc_beta_ddz (T a, T b, T z)
	Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to z. More...
template<>
double	inc_beta_ddz (double a, double b, double z)
template<typename T >
int	int_step (const T &y)
	The integer step, or Heaviside, function. More...
double	inv (double x)
double	inv_cloglog (double x)
	The inverse complementary log-log function. More...
double	inv_logit (double a)
	Returns the inverse logit function applied to the argument. More...
double	inv_Phi (double p)
	The inverse of the unit normal cumulative distribution function. More...
double	inv_sqrt (double x)
double	inv_square (double x)
template<typename T >
bool	is_any_nan (T &&x)
	Returns true if the input is NaN and false otherwise. More...
template<typename T , typename... Ts>
bool	is_any_nan (T &&x, Ts &&...xs)
	Returns true if any input is NaN and false otherwise. More...
bool	is_inf (double x)
	Returns true if the input is infinite and false otherwise. More...
template<typename T >
bool	is_integer (T x)
	Returns true if the input is an integer and false otherwise. More...
template<typename T , typename = require_arithmetic_t<T>>
bool	is_nan (T x)
	Returns true if the input is NaN and false otherwise. More...
template<typename T >
bool	is_nonpositive_integer (T x)
	Returns true if the input is a nonpositive integer and false otherwise. More...
template<typename T >
bool	is_uninitialized (T x)
	Returns true if the specified variable is uninitialized. More...
template<typename T , typename L >
return_type_t< T, L >	lb_constrain (const T &x, const L &lb)
	Return the lower-bounded value for the specified unconstrained input and specified lower bound. More...
template<typename T , typename L >
return_type_t< T, L >	lb_constrain (const T &x, const L &lb, T &lp)
	Return the lower-bounded value for the speicifed unconstrained input and specified lower bound, incrementing the specified reference with the log absolute Jacobian determinant of the transform. More...
template<typename T , typename L >
return_type_t< T, L >	lb_free (const T &y, const L &lb)
	Return the unconstrained value that produces the specified lower-bound constrained value. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	lbeta (const T1 a, const T2 b)
	Return the log of the beta function applied to the specified arguments. More...
template<typename T1 , typename = require_arithmetic_t<T1>>
double	ldexp (T1 a, int b)
	Returns the product of a (the significand) and 2 to power b (the exponent). More...
double	lgamma (double x)
	Return the natural logarithm of the gamma function applied to the specified argument. More...
double	lgamma (int x)
	Return the natural logarithm of the gamma function applied to the specified argument. More...
template<typename T >
return_type_t< T >	lmgamma (int k, T x)
	Return the natural logarithm of the multivariate gamma function with the speciifed dimensions and argument. More...
double	log (int x)
	Return the natural log of the specified argument. More...
double	log1m (double x)
	Return the natural logarithm of one minus the specified value. More...
double	log1m_exp (double a)
	Calculates the natural logarithm of one minus the exponential of the specified value without overflow,. More...
double	log1m_inv_logit (double u)
	Returns the natural logarithm of 1 minus the inverse logit of the specified argument. More...
double	log1m_inv_logit (int u)
	Return the natural logarithm of one minus the inverse logit of the specified argument. More...
double	log1p (double x)
	Return the natural logarithm of one plus the specified value. More...
double	log1p (int x)
	Return the natural logarithm of one plus the specified argument. More...
double	log1p_exp (double a)
	Calculates the log of 1 plus the exponential of the specified value without overflow. More...
template<typename T , typename = require_arithmetic_t<T>>
double	log2 (T u)
	Returns the base two logarithm of the argument (C99, C++11). More...
double	log2 ()
	Return natural logarithm of two. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	log_diff_exp (const T1 x, const T2 y)
	The natural logarithm of the difference of the natural exponentiation of x and the natural exponentiation of y. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	log_falling_factorial (const T1 x, const T2 n)
	Return the natural log of the falling factorial of the specified arguments. More...
double	log_inv_logit (double u)
	Returns the natural logarithm of the inverse logit of the specified argument. More...
double	log_inv_logit (int u)
	Returns the natural logarithm of the inverse logit of the specified argument. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	log_inv_logit_diff (const T1 &x, const T2 &y)
	Returns the natural logarithm of the difference of the inverse logits of the specified arguments. More...
double	log_mix (double theta, double lambda1, double lambda2)
	Return the log mixture density with specified mixing proportion and log densities. More...
double	log_mix (double theta, double lambda1, int lambda2)
	Return the log mixture density with specified mixing proportion and log densities. More...
double	log_mix (double theta, int lambda1, double lambda2)
	Return the log mixture density with specified mixing proportion and log densities. More...
double	log_mix (int theta, double lambda1, double lambda2)
	Return the log mixture density with specified mixing proportion and log densities. More...
double	log_mix (double theta, int lambda1, int lambda2)
	Return the log mixture density with specified mixing proportion and log densities. More...
double	log_mix (int theta, double lambda1, int lambda2)
	Return the log mixture density with specified mixing proportion and log densities. More...
double	log_mix (int theta, int lambda1, int lambda2)
	Return the log mixture density with specified mixing proportion and log densities. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2, double >	log_modified_bessel_first_kind (const T1 v, const T2 z)
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	log_rising_factorial (const T1 &x, const T2 &n)
	Return the natural logarithm of the rising factorial from the first specified argument to the second. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	log_sum_exp (const T2 &a, const T1 &b)
	Calculates the log sum of exponetials without overflow. More...
template<typename T1 , typename T2 >
int	logical_and (const T1 x1, const T2 x2)
	The logical and function which returns 1 if both arguments are unequal to zero and 0 otherwise. More...
template<typename T1 , typename T2 >
int	logical_eq (const T1 x1, const T2 x2)
	Return 1 if the first argument is equal to the second. More...
template<typename T1 , typename T2 >
bool	logical_gt (const T1 x1, const T2 x2)
	Return 1 if the first argument is strictly greater than the second. More...
template<typename T1 , typename T2 >
bool	logical_gte (const T1 x1, const T2 x2)
	Return 1 if the first argument is greater than or equal to the second. More...
template<typename T1 , typename T2 >
bool	logical_lt (T1 x1, T2 x2)
	Return 1 if the first argument is strictly less than the second. More...
template<typename T1 , typename T2 >
bool	logical_lte (const T1 x1, const T2 x2)
	Return 1 if the first argument is less than or equal to the second. More...
template<typename T >
int	logical_negation (const T &x)
	The logical negation function which returns one if the input is equal to zero and zero otherwise. More...
template<typename T1 , typename T2 >
int	logical_neq (const T1 x1, const T2 x2)
	Return 1 if the first argument is unequal to the second. More...
template<typename T1 , typename T2 >
int	logical_or (T1 x1, T2 x2)
	The logical or function which returns 1 if either argument is unequal to zero and 0 otherwise. More...
double	logit (double u)
	Return the log odds of the argument. More...
double	logit (int u)
	Return the log odds of the argument. More...
template<typename T , typename L , typename U >
return_type_t< T, L, U >	lub_constrain (const T &x, const L &lb, const U &ub)
	Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds. More...
template<typename T , typename L , typename U >
return_type_t< T, L, U >	lub_constrain (const T &x, const L &lb, const U &ub, T &lp)
	Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds and increment the specified log probability with the log absolute Jacobian determinant. More...
template<typename T , typename L , typename U >
return_type_t< T, L, U >	lub_free (const T &y, const L &lb, const U &ub)
	Return the unconstrained scalar that transforms to the specified lower- and upper-bounded scalar given the specified bounds. More...
template<typename T2 >
T2	modified_bessel_first_kind (int v, const T2 z)
	\[ \mbox{modified\_bessel\_first\_kind}(v, z) = \begin{cases} I_v(z) & \mbox{if } -\infty\leq z \leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \] More...
template<typename T2 >
T2	modified_bessel_second_kind (int v, const T2 z)
	\[ \mbox{modified\_bessel\_second\_kind}(v, z) = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ K_v(z) & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \] More...
int	modulus (int x, int y)
template<typename T_a , typename T_b >
return_type_t< T_a, T_b >	multiply_log (const T_a a, const T_b b)
	Calculated the value of the first argument times log of the second argument while behaving properly with 0 inputs. More...
template<typename T , typename M , typename S >
return_type_t< T, M, S >	offset_multiplier_constrain (const T &x, const M &mu, const S &sigma)
	Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier. More...
template<typename T , typename M , typename S >
return_type_t< T, M, S >	offset_multiplier_constrain (const T &x, const M &mu, const S &sigma, T &lp)
	Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier, incrementing the specified reference with the log absolute Jacobian determinant of the transform. More...
template<typename T , typename L , typename S >
return_type_t< T, L, S >	offset_multiplier_free (const T &y, const L &mu, const S &sigma)
	Return the unconstrained scalar that transforms to the specified offset and multiplier constrained scalar given the specified offset and multiplier. More...
double	owens_t (double h, double a)
	Return the result of applying Owen's T function to the specified arguments. More...
double	Phi (double x)
	The unit normal cumulative distribution function. More...
double	Phi_approx (double x)
	Return an approximation of the unit normal CDF. More...
double	Phi_approx (int x)
	Return an approximation of the unit normal CDF. More...
template<typename T >
T	positive_constrain (const T &x)
	Return the positive value for the specified unconstrained input. More...
template<typename T >
T	positive_constrain (const T &x, T &lp)
	Return the positive value for the specified unconstrained input, incrementing the scalar reference with the log absolute Jacobian determinant. More...
template<typename T >
T	positive_free (const T &y)
	Return the unconstrained value corresponding to the specified positive-constrained value. More...
template<typename T >
std::enable_if< std::is_arithmetic< T >::value, T >::type	primitive_value (T x)
	Return the value of the specified arithmetic argument unmodified with its own declared type. More...
template<typename T >
std::enable_if<!std::is_arithmetic< T >::value, double >::type	primitive_value (const T &x)
	Return the primitive value of the specified argument. More...
template<typename T >
T	prob_constrain (const T &x)
	Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar. More...
template<typename T >
T	prob_constrain (const T &x, T &lp)
	Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<typename T >
T	prob_free (const T &y)
	Return the free scalar that when transformed to a probability produces the specified scalar. More...
template<typename T , typename S >
promote_scalar_type< T, S >::type	promote_scalar (const S &x)
	This is the top-level function to call to promote the scalar types of an input of type S to type T. More...
template<typename T >
return_type_t< T >	rising_factorial (const T &x, int n)
	Return the rising factorial function evaluated at the inputs. More...
double	round (double x)
	Return the closest integer to the specified argument, with halfway cases rounded away from zero. More...
double	round (int x)
	Return the closest integer to the specified argument, with halfway cases rounded away from zero. More...
template<typename T >
int	sign (const T &z)
template<typename T >
bool	size_zero (T &x)
	Returns 1 if input is of length 0, returns 0 otherwise. More...
template<typename T , typename... Ts>
bool	size_zero (T &x, Ts &&...xs)
	Returns 1 if any inputs are of length 0, returns 0 otherwise. More...
double	sqrt (int x)
	Return the square root of the specified argument. More...
double	square (double x)
	Return the square of the specified argument. More...
template<typename T1 , typename T2 >
return_type_t< T1, T2 >	squared_distance (const T1 &x1, const T2 &x2)
	Returns the squared distance. More...
template<typename T >
double	step (const T &y)
	The step, or Heaviside, function. More...
double	sum (double v)
	Returns specified input value. More...
int	sum (int v)
	Returns specified input value. More...
double	tgamma (double x)
	Return the gamma function applied to the specified argument. More...
template<typename T >
T	trigamma_impl (const T &x)
	Return the trigamma function applied to the argument. More...
double	trigamma (double u)
	Return the second derivative of the log Gamma function evaluated at the specified argument. More...
double	trigamma (int u)
	Return the second derivative of the log Gamma function evaluated at the specified argument. More...
double	trunc (double x)
	Return the nearest integral value that is not larger in magnitude than the specified argument. More...
double	trunc (int x)
	Return the nearest integral value that is not larger in magnitude than the specified argument. More...
template<typename T , typename U >
return_type_t< T, U >	ub_constrain (const T &x, const U &ub)
	Return the upper-bounded value for the specified unconstrained scalar and upper bound. More...
template<typename T , typename U >
return_type_t< T, U >	ub_constrain (const T &x, const U &ub, T &lp)
	Return the upper-bounded value for the specified unconstrained scalar and upper bound and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
template<typename T , typename U >
return_type_t< T, U >	ub_free (const T &y, const U &ub)
	Return the free scalar that corresponds to the specified upper-bounded value with respect to the specified upper bound. More...
template<typename T >
double	value_of (const T x)
	Return the value of the specified scalar argument converted to a double value. More...
template<>
double	value_of< double > (double x)
	Return the specified argument. More...
int	value_of (int x)
	Return the specified argument. More...
template<typename T >
double	value_of_rec (const T x)
	Return the value of the specified scalar argument converted to a double value. More...
template<>
double	value_of_rec< double > (double x)
	Return the specified argument. More...
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_ccdf_log (const T_n &n, const T_prob &theta)
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_cdf (const T_n &n, const T_prob &theta)
	Returns the CDF of the Bernoulli distribution. More...
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_cdf_log (const T_n &n, const T_prob &theta)
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_lccdf (const T_n &n, const T_prob &theta)
	Returns the log CCDF of the Bernoulli distribution. More...
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_lcdf (const T_n &n, const T_prob &theta)
	Returns the log CDF of the Bernoulli distribution. More...
template<bool propto, typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_log (const T_n &n, const T_prob &theta)
template<typename T_y , typename T_prob >
return_type_t< T_prob >	bernoulli_log (const T_y &n, const T_prob &theta)
template<bool propto, typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_logit_log (const T_n &n, const T_prob &theta)
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_logit_log (const T_n &n, const T_prob &theta)
template<bool propto, typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_logit_lpmf (const T_n &n, const T_prob &theta)
	Returns the log PMF of the logit-parametrized Bernoulli distribution. More...
template<typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_logit_lpmf (const T_n &n, const T_prob &theta)
template<typename T_t , class RNG >
VectorBuilder< true, int, T_t >::type	bernoulli_logit_rng (const T_t &t, RNG &rng)
	Return a Bernoulli random variate with logit-parameterized chance of success using the specified random number generator. More...
template<bool propto, typename T_n , typename T_prob >
return_type_t< T_prob >	bernoulli_lpmf (const T_n &n, const T_prob &theta)
	Returns the log PMF of the Bernoulli distribution. More...
template<typename T_y , typename T_prob >
return_type_t< T_prob >	bernoulli_lpmf (const T_y &n, const T_prob &theta)
template<typename T_theta , class RNG >
VectorBuilder< true, int, T_theta >::type	bernoulli_rng (const T_theta &theta, RNG &rng)
	Return a Bernoulli random variate with specified chance of success parameter using the specified random number generator. More...
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_ccdf_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_cdf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
	Returns the CDF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_cdf_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_lccdf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
	Returns the log CCDF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_lcdf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
	Returns the log CDF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<bool propto, typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<bool propto, typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_lpmf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
	Returns the log PMF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters. More...
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type_t< T_size1, T_size2 >	beta_binomial_lpmf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
template<typename T_N , typename T_shape1 , typename T_shape2 , class RNG >
VectorBuilder< true, int, T_N, T_shape1, T_shape2 >::type	beta_binomial_rng (const T_N &N, const T_shape1 &alpha, const T_shape2 &beta, RNG &rng)
	Return a beta-binomial random variate with the specified population size, success, and failure parameters using the given random number generator. More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_ccdf_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_cdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
	Calculates the beta cumulative distribution function for the given variate and scale variables. More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_cdf_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_lccdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
	Returns the beta log complementary cumulative distribution function for the given probability, success, and failure parameters. More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_lcdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
	Returns the beta log cumulative distribution function for the given probability, success, and failure parameters. More...
template<bool propto, typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
	The log of the beta density for the specified scalar(s) given the specified sample size(s). More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<bool propto, typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_lpdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
	The log of the beta density for the specified scalar(s) given the specified sample size(s). More...
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type_t< T_y, T_scale_succ, T_scale_fail >	beta_lpdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_ccdf_log (const T_y &y, const T_loc &mu, const T_prec &kappa)
	Returns the beta log complementary cumulative distribution function for specified probability, location, and precision parameters: beta_proportion_lccdf(y | mu, kappa) = beta_lccdf(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_cdf_log (const T_y &y, const T_loc &mu, const T_prec &kappa)
	Returns the beta log cumulative distribution function for specified probability, location, and precision parameters: beta_proportion_lcdf(y | mu, kappa) = beta_lcdf(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_lccdf (const T_y &y, const T_loc &mu, const T_prec &kappa)
	Returns the beta log complementary cumulative distribution function for specified probability, location, and precision parameters: beta_proportion_lccdf(y | mu, kappa) = beta_lccdf(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_lcdf (const T_y &y, const T_loc &mu, const T_prec &kappa)
	Returns the beta log cumulative distribution function for specified probability, location, and precision parameters: beta_proportion_lcdf(y | mu, kappa) = beta_lcdf(y | mu * kappa, (1 - mu) * kappa). More...
template<bool propto, typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_log (const T_y &y, const T_loc &mu, const T_prec &kappa)
	The log of the beta density for specified y, location, and precision: beta_proportion_lpdf(y | mu, kappa) = beta_lpdf(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_log (const T_y &y, const T_loc &mu, const T_prec &kappa)
template<bool propto, typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_lpdf (const T_y &y, const T_loc &mu, const T_prec &kappa)
	The log of the beta density for specified y, location, and precision: beta_proportion_lpdf(y | mu, kappa) = beta_lpdf(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_y , typename T_loc , typename T_prec >
return_type_t< T_y, T_loc, T_prec >	beta_proportion_lpdf (const T_y &y, const T_loc &mu, const T_prec &kappa)
template<typename T_loc , typename T_prec , class RNG >
VectorBuilder< true, double, T_loc, T_prec >::type	beta_proportion_rng (const T_loc &mu, const T_prec &kappa, RNG &rng)
	Return a Beta random variate specified probability, location, and precision parameters: beta_proportion_rng(y | mu, kappa) = beta_rng(y | mu * kappa, (1 - mu) * kappa). More...
template<typename T_shape1 , typename T_shape2 , class RNG >
VectorBuilder< true, double, T_shape1, T_shape2 >::type	beta_rng (const T_shape1 &alpha, const T_shape2 &beta, RNG &rng)
	Return a Beta random variate with the supplied success and failure parameters using the given random number generator. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_ccdf_log (const T_n &n, const T_N &N, const T_prob &theta)
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_cdf (const T_n &n, const T_N &N, const T_prob &theta)
	Returns the CDF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_cdf_log (const T_n &n, const T_N &N, const T_prob &theta)
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_lccdf (const T_n &n, const T_N &N, const T_prob &theta)
	Returns the log CCDF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_lcdf (const T_n &n, const T_N &N, const T_prob &theta)
	Returns the log CDF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_log (const T_n &n, const T_N &N, const T_prob &theta)
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_log (const T_n &n, const T_N &N, const T_prob &theta)
template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_logit_log (const T_n &n, const T_N &N, const T_prob &alpha)
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_logit_log (const T_n &n, const T_N &N, const T_prob &alpha)
template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_logit_lpmf (const T_n &n, const T_N &N, const T_prob &alpha)
	Binomial log PMF in logit parametrization. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_logit_lpmf (const T_n &n, const T_N &N, const T_prob &alpha)
template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_lpmf (const T_n &n, const T_N &N, const T_prob &theta)
	Returns the log PMF for the binomial distribution evaluated at the specified success, population size, and chance of success. More...
template<typename T_n , typename T_N , typename T_prob >
return_type_t< T_prob >	binomial_lpmf (const T_n &n, const T_N &N, const T_prob &theta)
template<typename T_N , typename T_theta , class RNG >
VectorBuilder< true, int, T_N, T_theta >::type	binomial_rng (const T_N &N, const T_theta &theta, RNG &rng)
	Return a pseudorandom binomial random variable for the given population size and chance of success parameters using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the cauchy cumulative distribution function for the given location, and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the cauchy log complementary cumulative distribution function for the given location, and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the cauchy log cumulative distribution function for the given location, and scale. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
	The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s). More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s). More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	cauchy_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	cauchy_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a Cauchy random variate for the given location and scale using the specified random number generator. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_ccdf_log (const T_y &y, const T_dof &nu)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_cdf (const T_y &y, const T_dof &nu)
	Returns the chi square cumulative distribution function for the given variate and degrees of freedom. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_cdf_log (const T_y &y, const T_dof &nu)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_lccdf (const T_y &y, const T_dof &nu)
	Returns the chi square log complementary cumulative distribution function for the given variate and degrees of freedom. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_lcdf (const T_y &y, const T_dof &nu)
	Returns the chi square log cumulative distribution function for the given variate and degrees of freedom. More...
template<bool propto, typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_log (const T_y &y, const T_dof &nu)
	The log of a chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_log (const T_y &y, const T_dof &nu)
template<bool propto, typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_lpdf (const T_y &y, const T_dof &nu)
	The log of a chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	chi_square_lpdf (const T_y &y, const T_dof &nu)
template<typename T_deg , class RNG >
VectorBuilder< true, double, T_deg >::type	chi_square_rng (const T_deg &nu, RNG &rng)
	Return a chi squared random variate with nu degrees of freedom using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the double exponential cumulative density function. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the double exponential log complementary cumulative density function. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the double exponential log cumulative density function. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Returns the double exponential log probability density function. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	double_exponential_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	double_exponential_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a double exponential random variate with the given location and scale using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type_t< T_y, T_loc, T_scale, T_inv_scale >	exp_mod_normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
template<typename T_loc , typename T_scale , typename T_inv_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale, T_inv_scale >::type	exp_mod_normal_rng (const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda, RNG &rng)
	Return an exponentially modified normal random variate for the given location, scale, and inverse scale using the specified random number generator. More...
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_ccdf_log (const T_y &y, const T_inv_scale &beta)
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_cdf (const T_y &y, const T_inv_scale &beta)
	Calculates the exponential cumulative distribution function for the given y and beta. More...
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_cdf_log (const T_y &y, const T_inv_scale &beta)
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_lccdf (const T_y &y, const T_inv_scale &beta)
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_lcdf (const T_y &y, const T_inv_scale &beta)
template<bool propto, typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_log (const T_y &y, const T_inv_scale &beta)
	The log of an exponential density for y with the specified inverse scale parameter. More...
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_log (const T_y &y, const T_inv_scale &beta)
template<bool propto, typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_lpdf (const T_y &y, const T_inv_scale &beta)
	The log of an exponential density for y with the specified inverse scale parameter. More...
template<typename T_y , typename T_inv_scale >
return_type_t< T_y, T_inv_scale >	exponential_lpdf (const T_y &y, const T_inv_scale &beta)
template<typename T_inv , class RNG >
VectorBuilder< true, double, T_inv >::type	exponential_rng (const T_inv &beta, RNG &rng)
	Return a exponential random variate with inverse scale beta using the specified random number generator. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_cdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_lccdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_lcdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_lpdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	frechet_lpdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_shape , typename T_scale , class RNG >
VectorBuilder< true, double, T_shape, T_scale >::type	frechet_rng (const T_shape &alpha, const T_scale &sigma, RNG &rng)
	Return a pseudorandom Frechet variate for the given shape and scale parameters using the specified random number generator. More...
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_ccdf_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_cdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
	The cumulative density function for a gamma distribution for y with the specified shape and inverse scale parameters. More...
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_cdf_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_lccdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_lcdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<bool propto, typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
	The log of a gamma density for y with the specified shape and inverse scale parameters. More...
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<bool propto, typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_lpdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
	The log of a gamma density for y with the specified shape and inverse scale parameters. More...
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type_t< T_y, T_shape, T_inv_scale >	gamma_lpdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_shape , typename T_inv , class RNG >
VectorBuilder< true, double, T_shape, T_inv >::type	gamma_rng (const T_shape &alpha, const T_inv &beta, RNG &rng)
	Return a gamma random variate for the given shape and inverse scale parameters using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &beta)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_cdf (const T_y &y, const T_loc &mu, const T_scale &beta)
	Returns the Gumbel distribution cumulative distribution for the given location and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_cdf_log (const T_y &y, const T_loc &mu, const T_scale &beta)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_lccdf (const T_y &y, const T_loc &mu, const T_scale &beta)
	Returns the Gumbel log complementary cumulative distribution for the given location and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_lcdf (const T_y &y, const T_loc &mu, const T_scale &beta)
	Returns the Gumbel log cumulative distribution for the given location and scale. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_log (const T_y &y, const T_loc &mu, const T_scale &beta)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_log (const T_y &y, const T_loc &mu, const T_scale &beta)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_lpdf (const T_y &y, const T_loc &mu, const T_scale &beta)
	Returns the Gumbel log probability density for the given location and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	gumbel_lpdf (const T_y &y, const T_loc &mu, const T_scale &beta)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	gumbel_rng (const T_loc &mu, const T_scale &beta, RNG &rng)
	Return a Gumbel random variate with the given location and scale using the specified random number generator. More...
template<bool propto, typename T_n , typename T_N , typename T_a , typename T_b >
double	hypergeometric_log (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
template<typename T_n , typename T_N , typename T_a , typename T_b >
double	hypergeometric_log (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
template<bool propto, typename T_n , typename T_N , typename T_a , typename T_b >
double	hypergeometric_lpmf (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
template<typename T_n , typename T_N , typename T_a , typename T_b >
double	hypergeometric_lpmf (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
template<class RNG >
int	hypergeometric_rng (int N, int a, int b, RNG &rng)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_ccdf_log (const T_y &y, const T_dof &nu)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_cdf (const T_y &y, const T_dof &nu)
	Returns the inverse chi square cumulative distribution function for the given variate and degrees of freedom. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_cdf_log (const T_y &y, const T_dof &nu)
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_lccdf (const T_y &y, const T_dof &nu)
	Returns the inverse chi square log complementary cumulative distribution function for the given variate and degrees of freedom. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_lcdf (const T_y &y, const T_dof &nu)
	Returns the inverse chi square log cumulative distribution function for the given variate and degrees of freedom. More...
template<bool propto, typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_log (const T_y &y, const T_dof &nu)
	The log of an inverse chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_log (const T_y &y, const T_dof &nu)
template<bool propto, typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_lpdf (const T_y &y, const T_dof &nu)
	The log of an inverse chi-squared density for y with the specified degrees of freedom parameter. More...
template<typename T_y , typename T_dof >
return_type_t< T_y, T_dof >	inv_chi_square_lpdf (const T_y &y, const T_dof &nu)
template<typename T_deg , class RNG >
VectorBuilder< true, double, T_deg >::type	inv_chi_square_rng (const T_deg &nu, RNG &rng)
	Return a pseudorandom inverse chi squared variate with the nu degrees of freedom using the specified random number generator. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_cdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
	The CDF of an inverse gamma density for y with the specified shape and scale parameters. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_lccdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_lcdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
	The log of an inverse gamma density for y with the specified shape and scale parameters. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_lpdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
	The log of an inverse gamma density for y with the specified shape and scale parameters. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	inv_gamma_lpdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
template<typename T_shape , typename T_scale , class RNG >
VectorBuilder< true, double, T_shape, T_scale >::type	inv_gamma_rng (const T_shape &alpha, const T_scale &beta, RNG &rng)
	Return a pseudorandom inverse gamma variate for the given shape and scale parameters using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	logistic_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	logistic_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a Logistic random variate for the given location and scale using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	lognormal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	lognormal_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a lognormal random variate for the given location and scale using the specified random number generator. More...
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_ccdf_log (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_cdf (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_cdf_log (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_lccdf (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_lcdf (const T_n &n, const T_location &mu, const T_precision &phi)
template<bool propto, typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_log (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_log (const T_n &n, const T_location &mu, const T_precision &phi)
template<bool propto, typename T_n , typename T_log_location , typename T_precision >
return_type_t< T_log_location, T_precision >	neg_binomial_2_log_log (const T_n &n, const T_log_location &eta, const T_precision &phi)
template<typename T_n , typename T_log_location , typename T_precision >
return_type_t< T_log_location, T_precision >	neg_binomial_2_log_log (const T_n &n, const T_log_location &eta, const T_precision &phi)
template<bool propto, typename T_n , typename T_log_location , typename T_precision >
return_type_t< T_log_location, T_precision >	neg_binomial_2_log_lpmf (const T_n &n, const T_log_location &eta, const T_precision &phi)
template<typename T_n , typename T_log_location , typename T_precision >
return_type_t< T_log_location, T_precision >	neg_binomial_2_log_lpmf (const T_n &n, const T_log_location &eta, const T_precision &phi)
template<typename T_loc , typename T_inv , class RNG >
VectorBuilder< true, int, T_loc, T_inv >::type	neg_binomial_2_log_rng (const T_loc &eta, const T_inv &phi, RNG &rng)
	Return a negative binomial random variate with the specified log-location and inverse dispersion parameters using the given random number generator. More...
template<bool propto, typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_lpmf (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_n , typename T_location , typename T_precision >
return_type_t< T_location, T_precision >	neg_binomial_2_lpmf (const T_n &n, const T_location &mu, const T_precision &phi)
template<typename T_loc , typename T_prec , class RNG >
VectorBuilder< true, int, T_loc, T_prec >::type	neg_binomial_2_rng (const T_loc &mu, const T_prec &phi, RNG &rng)
	Return a negative binomial random variate with the specified location and precision parameters using the given random number generator. More...
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_ccdf_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_cdf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_cdf_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_lccdf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_lcdf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<bool propto, typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<bool propto, typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_lpmf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type_t< T_shape, T_inv_scale >	neg_binomial_lpmf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
template<typename T_shape , typename T_inv , class RNG >
VectorBuilder< true, int, T_shape, T_inv >::type	neg_binomial_rng (const T_shape &alpha, const T_inv &beta, RNG &rng)
	Return a negative binomial random variate with the specified shape and inverse scale parameters using the given random number generator. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	Calculates the normal cumulative distribution function for the given variate, location, and scale. More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
	The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s). More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
	The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s). More...
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
template<typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_loc, T_scale >::type	normal_rng (const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a Normal random variate for the given location and scale using the specified random number generator. More...
template<bool propto, typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >
return_type_t< T_y, T_s, T_loc, T_scale >	normal_sufficient_log (const T_y &y_bar, const T_s &s_squared, const T_n &n_obs, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >
return_type_t< T_y, T_s, T_loc, T_scale >	normal_sufficient_log (const T_y &y_bar, const T_s &s_squared, const T_n &n_obs, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >
return_type_t< T_y, T_s, T_loc, T_scale >	normal_sufficient_lpdf (const T_y &y_bar, const T_s &s_squared, const T_n &n_obs, const T_loc &mu, const T_scale &sigma)
	The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s). More...
template<typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >
return_type_t< T_y, T_s, T_loc, T_scale >	normal_sufficient_lpdf (const T_y &y_bar, const T_s &s_squared, const T_n &n_obs, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_ccdf_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_cdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_cdf_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_lccdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_lcdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<bool propto, typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<bool propto, typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_lpdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_y , typename T_scale , typename T_shape >
return_type_t< T_y, T_scale, T_shape >	pareto_lpdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
template<typename T_shape , typename T_scale , class RNG >
VectorBuilder< true, double, T_shape, T_scale >::type	pareto_rng (const T_scale &y_min, const T_shape &alpha, RNG &rng)
	Return a Pareto random variate for the given shape and scale parameters using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_cdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_cdf_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_lccdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_lcdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_lpdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	pareto_type_2_lpdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
template<typename T_loc , typename T_scale , typename T_shape , class RNG >
VectorBuilder< true, double, T_loc, T_scale, T_shape >::type	pareto_type_2_rng (const T_loc &mu, const T_scale &lambda, const T_shape &alpha, RNG &rng)
	Return a Pareto type 2 random variate for the given location, scale, and shape using the specified random number generator. More...
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_ccdf_log (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_cdf (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_cdf_log (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_lccdf (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_lcdf (const T_n &n, const T_rate &lambda)
template<bool propto, typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_log (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_log (const T_n &n, const T_rate &lambda)
template<bool propto, typename T_n , typename T_log_rate >
return_type_t< T_log_rate >	poisson_log_log (const T_n &n, const T_log_rate &alpha)
template<typename T_n , typename T_log_rate >
return_type_t< T_log_rate >	poisson_log_log (const T_n &n, const T_log_rate &alpha)
template<bool propto, typename T_n , typename T_log_rate >
return_type_t< T_log_rate >	poisson_log_lpmf (const T_n &n, const T_log_rate &alpha)
template<typename T_n , typename T_log_rate >
return_type_t< T_log_rate >	poisson_log_lpmf (const T_n &n, const T_log_rate &alpha)
template<typename T_rate , class RNG >
VectorBuilder< true, int, T_rate >::type	poisson_log_rng (const T_rate &alpha, RNG &rng)
	Return a Poisson random variate with specified log rate parameter using the given random number generator. More...
template<bool propto, typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_lpmf (const T_n &n, const T_rate &lambda)
template<typename T_n , typename T_rate >
return_type_t< T_rate >	poisson_lpmf (const T_n &n, const T_rate &lambda)
template<typename T_rate , class RNG >
VectorBuilder< true, int, T_rate >::type	poisson_rng (const T_rate &lambda, RNG &rng)
	Return a Poisson random variate with specified rate parameter using the given random number generator. More...
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_ccdf_log (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_cdf (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_cdf_log (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_lccdf (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_lcdf (const T_y &y, const T_scale &sigma)
template<bool propto, typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_log (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_log (const T_y &y, const T_scale &sigma)
template<bool propto, typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_lpdf (const T_y &y, const T_scale &sigma)
template<typename T_y , typename T_scale >
return_type_t< T_y, T_scale >	rayleigh_lpdf (const T_y &y, const T_scale &sigma)
template<typename T_scale , class RNG >
VectorBuilder< true, double, T_scale >::type	rayleigh_rng (const T_scale &sigma, RNG &rng)
	Return a Rayleigh random variate with scale parameter sigma using the specified random number generator. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_ccdf_log (const T_y &y, const T_dof &nu, const T_scale &s)
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_cdf (const T_y &y, const T_dof &nu, const T_scale &s)
	The CDF of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_cdf_log (const T_y &y, const T_dof &nu, const T_scale &s)
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_lccdf (const T_y &y, const T_dof &nu, const T_scale &s)
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_lcdf (const T_y &y, const T_dof &nu, const T_scale &s)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_log (const T_y &y, const T_dof &nu, const T_scale &s)
	The log of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_log (const T_y &y, const T_dof &nu, const T_scale &s)
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_lpdf (const T_y &y, const T_dof &nu, const T_scale &s)
	The log of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
template<typename T_y , typename T_dof , typename T_scale >
return_type_t< T_y, T_dof, T_scale >	scaled_inv_chi_square_lpdf (const T_y &y, const T_dof &nu, const T_scale &s)
template<typename T_deg , typename T_scale , class RNG >
VectorBuilder< true, double, T_deg, T_scale >::type	scaled_inv_chi_square_rng (const T_deg &nu, const T_scale &s, RNG &rng)
	Return a scaled chi square random variate for the given number of degrees of freedom and scale using the specified random number generator. More...
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_lccdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_lcdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type_t< T_y, T_loc, T_scale, T_shape >	skew_normal_lpdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
template<typename T_loc , typename T_scale , typename T_shape , class RNG >
VectorBuilder< true, double, T_loc, T_scale, T_shape >::type	skew_normal_rng (const T_loc &mu, const T_scale &sigma, const T_shape &alpha, RNG &rng)
	Return a Skew-normal random variate for the given location, scale, and shape using the specified random number generator. More...
template<bool propto, typename T_y >
return_type_t< T_y >	std_normal_log (const T_y &y)
	The log of a standard normal density for the specified scalar(s). More...
template<typename T_y >
return_type_t< T_y >	std_normal_log (const T_y &y)
template<bool propto, typename T_y >
return_type_t< T_y >	std_normal_lpdf (const T_y &y)
	The log of the normal density for the specified scalar(s) given a location of 0 and a scale of 1. More...
template<typename T_y >
return_type_t< T_y >	std_normal_lpdf (const T_y &y)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_ccdf_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_cdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_cdf_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_lccdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_lcdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
	The log of the Student-t density for the given y, nu, mean, and scale parameter. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_lpdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
	The log of the Student-t density for the given y, nu, mean, and scale parameter. More...
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type_t< T_y, T_dof, T_loc, T_scale >	student_t_lpdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
template<typename T_deg , typename T_loc , typename T_scale , class RNG >
VectorBuilder< true, double, T_deg, T_loc, T_scale >::type	student_t_rng (const T_deg &nu, const T_loc &mu, const T_scale &sigma, RNG &rng)
	Return a student-t random variate for the given degrees of freedom, location, and scale using the specified random number generator. More...
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_ccdf_log (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_cdf (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_cdf_log (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_lccdf (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_lcdf (const T_y &y, const T_low &alpha, const T_high &beta)
template<bool propto, typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_log (const T_y &y, const T_low &alpha, const T_high &beta)
	The log of a uniform density for the given y, lower, and upper bound. More...
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_log (const T_y &y, const T_low &alpha, const T_high &beta)
template<bool propto, typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_lpdf (const T_y &y, const T_low &alpha, const T_high &beta)
	The log of a uniform density for the given y, lower, and upper bound. More...
template<typename T_y , typename T_low , typename T_high >
return_type_t< T_y, T_low, T_high >	uniform_lpdf (const T_y &y, const T_low &alpha, const T_high &beta)
template<typename T_alpha , typename T_beta , class RNG >
VectorBuilder< true, double, T_alpha, T_beta >::type	uniform_rng (const T_alpha &alpha, const T_beta &beta, RNG &rng)
	Return a uniform random variate for the given upper and lower bounds using the specified random number generator. More...
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	von_mises_log (T_y const &y, T_loc const &mu, T_scale const &kappa)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	von_mises_log (T_y const &y, T_loc const &mu, T_scale const &kappa)
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	von_mises_lpdf (T_y const &y, T_loc const &mu, T_scale const &kappa)
template<typename T_y , typename T_loc , typename T_scale >
return_type_t< T_y, T_loc, T_scale >	von_mises_lpdf (T_y const &y, T_loc const &mu, T_scale const &kappa)
template<typename T_loc , typename T_conc , class RNG >
VectorBuilder< true, double, T_loc, T_conc >::type	von_mises_rng (const T_loc &mu, const T_conc &kappa, RNG &rng)
	Return a von Mises random variate for the given location and concentration using the specified random number generator. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_cdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
	Returns the Weibull cumulative distribution function for the given location and scale. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_lccdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
	Returns the Weibull log complementary cumulative distribution function for the given location and scale. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_lcdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
	Returns the Weibull log cumulative distribution function for the given location and scale. More...
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_lpdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
	Returns the Weibull log probability density for the given location and scale. More...
template<typename T_y , typename T_shape , typename T_scale >
return_type_t< T_y, T_shape, T_scale >	weibull_lpdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
template<typename T_shape , typename T_scale , class RNG >
VectorBuilder< true, double, T_shape, T_scale >::type	weibull_rng (const T_shape &alpha, const T_scale &sigma, RNG &rng)
	Return a Weibull random variate for the given shape and scale parameters using the specified random number generator. More...
template<bool propto, typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type_t< T_y, T_alpha, T_tau, T_beta, T_delta >	wiener_log (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
	The log of the first passage time density function for a (Wiener) drift diffusion model for the given \(y\), boundary separation \(\alpha\), nondecision time \(\tau\), relative bias \(\beta\), and drift rate \(\delta\). More...
template<typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type_t< T_y, T_alpha, T_tau, T_beta, T_delta >	wiener_log (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
template<bool propto, typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type_t< T_y, T_alpha, T_tau, T_beta, T_delta >	wiener_lpdf (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
	The log of the first passage time density function for a (Wiener) drift diffusion model for the given \(y\), boundary separation \(\alpha\), nondecision time \(\tau\), relative bias \(\beta\), and drift rate \(\delta\). More...
template<typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type_t< T_y, T_alpha, T_tau, T_beta, T_delta >	wiener_lpdf (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
var	log_sum_exp (const std::vector< var > &x)
	Returns the log sum of exponentials. More...
var	sum (const std::vector< var > &m)
	Returns the sum of the entries of the specified vector. More...
std::vector< var >	to_var (const std::vector< double > &v)
	Converts argument to an automatic differentiation variable. More...
const std::vector< var > &	to_var (const std::vector< var > &v)
	Specialization of to_var to for const input vector of var. More...
std::vector< var > &	to_var (std::vector< var > &v)
	Specialization of to_var to for non-const input vector of var. More...
template<typename F >
double	gradient_of_f (const F &f, const double &x, const double &xc, const std::vector< double > &theta_vals, const std::vector< double > &x_r, const std::vector< int > &x_i, size_t n, std::ostream *msgs)
	Calculate first derivative of f(x, param, std::ostream&) with respect to the nth parameter. More...
template<typename F , typename T_a , typename T_b , typename T_theta , typename = require_any_var_t<T_a, T_b, T_theta>>
return_type_t< T_a, T_b, T_theta >	integrate_1d (const F &f, const T_a &a, const T_b &b, const std::vector< T_theta > &theta, const std::vector< double > &x_r, const std::vector< int > &x_i, std::ostream *msgs, const double relative_tolerance=std::sqrt(EPSILON))
	Compute the integral of the single variable function f from a to b to within a specified relative tolerance. More...
template<int R, int C>
vari **	build_vari_array (const Eigen::Matrix< var, R, C > &x)
	Allocates and populates a flat array of vari pointers in the autodiff arena with the varis pointed to by the vars in the input Eigen matrix. More...
static bool	empty_nested ()
	Return true if there is no nested autodiff being executed. More...
static void	grad (vari *vi)
	Compute the gradient for all variables starting from the specified root variable implementation. More...
static size_t	nested_size ()
var	operator+ (const var &a, const var &b)
	Addition operator for variables (C++). More...
var	operator+ (const var &a, double b)
	Addition operator for variable and scalar (C++). More...
var	operator+ (double a, const var &b)
	Addition operator for scalar and variable (C++). More...
var	operator/ (const var &a, const var &b)
	Division operator for two variables (C++). More...
var	operator/ (const var &a, double b)
	Division operator for dividing a variable by a scalar (C++). More...
var	operator/ (double a, const var &b)
	Division operator for dividing a scalar by a variable (C++). More...
bool	operator== (const var &a, const var &b)
	Equality operator comparing two variables' values (C++). More...
bool	operator== (const var &a, double b)
	Equality operator comparing a variable's value and a double (C++). More...
bool	operator== (double a, const var &b)
	Equality operator comparing a scalar and a variable's value (C++). More...
bool	operator> (const var &a, const var &b)
	Greater than operator comparing variables' values (C++). More...
bool	operator> (const var &a, double b)
	Greater than operator comparing variable's value and double (C++). More...
bool	operator> (double a, const var &b)
	Greater than operator comparing a double and a variable's value (C++). More...
bool	operator>= (const var &a, const var &b)
	Greater than or equal operator comparing two variables' values (C++). More...
bool	operator>= (const var &a, double b)
	Greater than or equal operator comparing variable's value and double (C++). More...
bool	operator>= (double a, const var &b)
	Greater than or equal operator comparing double and variable's value (C++). More...
bool	operator< (const var &a, const var &b)
	Less than operator comparing variables' values (C++). More...
bool	operator< (const var &a, double b)
	Less than operator comparing variable's value and a double (C++). More...
bool	operator< (double a, const var &b)
	Less than operator comparing a double and variable's value (C++). More...
bool	operator<= (const var &a, const var &b)
	Less than or equal operator comparing two variables' values (C++). More...
bool	operator<= (const var &a, double b)
	Less than or equal operator comparing a variable's value and a scalar (C++). More...
bool	operator<= (double a, const var &b)
	Less than or equal operator comparing a double and variable's value (C++). More...
bool	operator&& (const var &x, const var &y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename T >
bool	operator&& (const var &x, double y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
template<typename T >
bool	operator&& (double x, const var &y)
	Return the logical conjunction of the values of the two arguments as defined by &&. More...
bool	operator|| (const var &x, const var &y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename T >
bool	operator|| (const var &x, double y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
template<typename T >
bool	operator|| (double x, const var &y)
	Return the logical disjunction of the values of the two arguments as defined by ||. More...
var	operator* (const var &a, const var &b)
	Multiplication operator for two variables (C++). More...
var	operator* (const var &a, double b)
	Multiplication operator for a variable and a scalar (C++). More...
var	operator* (double a, const var &b)
	Multiplication operator for a scalar and a variable (C++). More...
bool	operator!= (const var &a, const var &b)
	Inequality operator comparing two variables' values (C++). More...
bool	operator!= (const var &a, double b)
	Inequality operator comparing a variable's value and a double (C++). More...
bool	operator!= (double a, const var &b)
	Inequality operator comparing a double and a variable's value (C++). More...
var	operator- (const var &a, const var &b)
	Subtraction operator for variables (C++). More...
var	operator- (const var &a, double b)
	Subtraction operator for variable and scalar (C++). More...
var	operator- (double a, const var &b)
	Subtraction operator for scalar and variable (C++). More...
var &	operator-- (var &a)
	Prefix decrement operator for variables (C++). More...
var	operator-- (var &a, int)
	Postfix decrement operator for variables (C++). More...
var &	operator++ (var &a)
	Prefix increment operator for variables (C++). More...
var	operator++ (var &a, int)
	Postfix increment operator for variables (C++). More...
var	operator- (const var &a)
	Unary negation operator for variables (C++). More...
bool	operator! (const var &x)
	Return the negation of the value of the argument as defined by !. More...
var	operator+ (const var &a)
	Unary plus operator for variables (C++). More...
var	precomputed_gradients (double value, const std::vector< var > &operands, const std::vector< double > &gradients)
	This function returns a var for an expression that has the specified value, vector of operands, and vector of partial derivatives of value with respect to the operands. More...
void	print_stack (std::ostream &o)
	Prints the auto-dif variable stack. More...
static void	recover_memory ()
	Recover memory used for all variables for reuse. More...
static void	recover_memory_nested ()
	Recover only the memory used for the top nested call. More...
static void	set_zero_all_adjoints ()
	Reset all adjoint values in the stack to zero. More...
static void	set_zero_all_adjoints_nested ()
	Reset all adjoint values in the top nested portion of the stack to zero. More...
static void	start_nested ()
	Record the current position so that recover_memory_nested() can find it. More...
static void	grad (vari *vi)
Eigen::Matrix< var,-1,-1 >	cholesky_decompose (const Eigen::Matrix< var,-1,-1 > &A)
	Reverse mode specialization of cholesky decomposition. More...
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename = require_any_var_t<T1, T2>>
Eigen::Matrix< return_type_t< T1, T2 >, 1, C1 >	columns_dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
template<int R, int C>
Eigen::Matrix< var, 1, C >	columns_dot_self (const Eigen::Matrix< var, R, C > &x)
	Returns the dot product of each column of a matrix with itself. More...
template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>
Eigen::Matrix< var,-1,-1 >	cov_exp_quad (const std::vector< T_x > &x, const var &sigma, const var &l)
template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>
Eigen::Matrix< var,-1,-1 >	cov_exp_quad (const std::vector< T_x > &x, double sigma, const var &l)
matrix_v	crossprod (const matrix_v &M)
	Returns the result of pre-multiplying a matrix by its own transpose. More...
template<int R, int C>
var	determinant (const Eigen::Matrix< var, R, C > &m)
template<int R, int C>
Eigen::Matrix< var, R, C >	divide (const Eigen::Matrix< double, R, C > &m, const var &c)
	Return matrix divided by scalar. More...
template<int R, int C>
Eigen::Matrix< var, R, C >	divide (const Eigen::Matrix< var, R, C > &m, const double &c)
	Return matrix divided by scalar. More...
template<int R, int C>
Eigen::Matrix< var, R, C >	divide (const Eigen::Matrix< var, R, C > &m, const var &c)
	Return matrix divided by scalar. More...
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename = require_any_var_t<T1, T2>>
return_type_t< T1, T2 >	dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
	Returns the dot product. More...
template<typename T1 , typename T2 , typename = require_any_var_t<T1, T2>>
return_type_t< T1, T2 >	dot_product (const T1 *v1, const T2 *v2, size_t length)
	Returns the dot product. More...
template<typename T1 , typename T2 , typename = require_any_var_t<T1, T2>>
return_type_t< T1, T2 >	dot_product (const std::vector< T1 > &v1, const std::vector< T2 > &v2)
	Returns the dot product. More...
template<int R, int C>
var	dot_self (const Eigen::Matrix< var, R, C > &v)
	Returns the dot product of a vector with itself. More...
template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>
Eigen::Matrix< var,-1,-1 >	gp_exp_quad_cov (const std::vector< T_x > &x, const var &sigma, const var &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>
Eigen::Matrix< var,-1,-1 >	gp_exp_quad_cov (const std::vector< T_x > &x, double sigma, const var &length_scale)
	Returns a squared exponential kernel. More...
template<typename T_x >
std::enable_if< std::is_same< typename scalar_type< T_x >::type, double >::value, Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic > >::type	gp_periodic_cov (const std::vector< T_x > &x, const var &sigma, const var &l, const var &p)
	Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \). More...
template<typename T_x >
std::enable_if< std::is_same< typename scalar_type< T_x >::type, double >::value, Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic > >::type	gp_periodic_cov (const std::vector< T_x > &x, double sigma, const var &l, const var &p)
	Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \). More...
void	grad (var &v, Eigen::Matrix< var, Eigen::Dynamic, 1 > &x, Eigen::VectorXd &g)
	Propagate chain rule to calculate gradients starting from the specified variable. More...
void	initialize_variable (var &variable, const var &value)
	Initialize variable to value. More...
template<int R, int C>
void	initialize_variable (Eigen::Matrix< var, R, C > &matrix, const var &value)
	Initialize every cell in the matrix to the specified value. More...
template<typename T >
void	initialize_variable (std::vector< T > &variables, const var &value)
	Initialize the variables in the standard vector recursively. More...
matrix_v	inverse (const matrix_v &m)
	Reverse mode specialization of calculating the inverse of the matrix. More...
template<int R, int C>
var	log_determinant (const Eigen::Matrix< var, R, C > &m)
template<int R, int C>
var	log_determinant_ldlt (LDLT_factor< var, R, C > &A)
template<int R, int C>
var	log_determinant_spd (const Eigen::Matrix< var, R, C > &m)
Eigen::Matrix< var, Eigen::Dynamic, 1 >	log_softmax (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &alpha)
	Return the softmax of the specified Eigen vector. More...
template<int R, int C>
var	log_sum_exp (const Eigen::Matrix< var, R, C > &x)
	Returns the log sum of exponentials. More...
template<typename Ta , typename Tb , int Cb>
Eigen::Matrix< typename stan::return_type< Ta, Tb >::type,-1, Cb >	matrix_exp_multiply (const Eigen::Matrix< Ta,-1,-1 > &A, const Eigen::Matrix< Tb,-1, Cb > &B)
	Wrapper of matrix_exp_action function for a more literal name. More...
template<int R, int C>
Eigen::Matrix< var, R, C >	matrix_power (const Eigen::Matrix< var, R, C > &M, const int n)
	Returns the nth power of the specific matrix. More...
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_ldlt (const LDLT_factor< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_ldlt (const LDLT_factor< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_ldlt (const LDLT_factor< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_spd (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_spd (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_spd (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_tri (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_tri (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_tri (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
template<typename T1 , typename T2 , typename = require_all_var_or_arithmetic_t<T1, T2>, typename = require_any_var_t<T1, T2>>
return_type_t< T1, T2 >	multiply (const T1 &v, const T2 &c)
	Return the product of two scalars. More...
template<int R, int C>
Eigen::Matrix< var, R, C >	multiply (const var &c, const Eigen::Matrix< var, R, C > &m)
	Return the product of scalar and matrix. More...
template<typename Arith , int R, int C, typename = require_arithmetic_t<Arith>>
Eigen::Matrix< var, R, C >	multiply (const Arith &c, const Eigen::Matrix< var, R, C > &m)
	Return the product of scalar and matrix. More...
template<typename Arith , int R, int C, typename = require_arithmetic_t<Arith>>
Eigen::Matrix< var, R, C >	multiply (const var &c, const Eigen::Matrix< Arith, R, C > &m)
	Return the product of scalar and matrix. More...
template<int R, int C>
Eigen::Matrix< var, R, C >	multiply (const Eigen::Matrix< var, R, C > &m, const var &c)
	Return the product of matrix and scalar. More...
template<typename Arith , int R, int C, typename = require_arithmetic_t<Arith>>
Eigen::Matrix< var, R, C >	multiply (const Eigen::Matrix< Arith, R, C > &m, const var &c)
	Return the product of matrix and scalar. More...
template<typename Arith , int R, int C, typename = require_arithmetic_t<Arith>>
Eigen::Matrix< var, R, C >	multiply (const Eigen::Matrix< var, R, C > &m, const Arith &c)
	Return the product of matrix and scalar. More...
template<typename Ta , int Ra, int Ca, typename Tb , int Cb, typename = require_any_var_t<Ta, Tb>>
Eigen::Matrix< var, Ra, Cb >	multiply (const Eigen::Matrix< Ta, Ra, Ca > &A, const Eigen::Matrix< Tb, Ca, Cb > &B)
	Return the product of two matrices. More...
template<typename Ta , int Ca, typename Tb , typename = require_any_var_t<Ta, Tb>>
var	multiply (const Eigen::Matrix< Ta, 1, Ca > &A, const Eigen::Matrix< Tb, Ca, 1 > &B)
	Return the scalar product of a row vector and a vector. More...
matrix_v	multiply_lower_tri_self_transpose (const matrix_v &L)
Eigen::Matrix< var, Eigen::Dynamic, 1 >	ordered_constrain (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &x)
	Return an increasing ordered vector derived from the specified free vector. More...
Eigen::Matrix< var, Eigen::Dynamic, 1 >	positive_ordered_constrain (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &x)
	Return an increasing positive ordered vector derived from the specified free vector. More...
template<typename Ta , int Ra, int Ca, typename Tb , int Rb, int Cb>
std::enable_if< std::is_same< Ta, var >::value||std::is_same< Tb, var >::value, Eigen::Matrix< var, Cb, Cb > >::type	quad_form (const Eigen::Matrix< Ta, Ra, Ca > &A, const Eigen::Matrix< Tb, Rb, Cb > &B)
template<typename Ta , int Ra, int Ca, typename Tb , int Rb>
std::enable_if< std::is_same< Ta, var >::value||std::is_same< Tb, var >::value, var >::type	quad_form (const Eigen::Matrix< Ta, Ra, Ca > &A, const Eigen::Matrix< Tb, Rb, 1 > &B)
template<typename Ta , int Ra, int Ca, typename Tb , int Rb, int Cb, require_any_var_t< Ta, Tb > ...>
Eigen::Matrix< var, Cb, Cb >	quad_form_sym (const Eigen::Matrix< Ta, Ra, Ca > &A, const Eigen::Matrix< Tb, Rb, Cb > &B)
template<typename Ta , int Ra, int Ca, typename Tb , int Rb, require_any_var_t< Ta, Tb > ...>
var	quad_form_sym (const Eigen::Matrix< Ta, Ra, Ca > &A, const Eigen::Matrix< Tb, Rb, 1 > &B)
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, require_any_var_t< T1, T2 > ...>
Eigen::Matrix< var, R1, 1 >	rows_dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
var	sd (const std::vector< var > &v)
	Return the sample standard deviation of the specified standard vector. More...
template<int R, int C>
var	sd (const Eigen::Matrix< var, R, C > &m)
Eigen::Matrix< var, Eigen::Dynamic, 1 >	simplex_constrain (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &y)
	Return the simplex corresponding to the specified free vector. More...
Eigen::Matrix< var, Eigen::Dynamic, 1 >	softmax (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &alpha)
	Return the softmax of the specified Eigen vector. More...
template<int R1, int C1, int R2, int C2>
var	squared_distance (const Eigen::Matrix< var, R1, C1 > &v1, const Eigen::Matrix< var, R2, C2 > &v2)
template<int R1, int C1, int R2, int C2>
var	squared_distance (const Eigen::Matrix< var, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
template<int R1, int C1, int R2, int C2>
var	squared_distance (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< var, R2, C2 > &v2)
void	stan_print (std::ostream *o, const var &x)
template<int R, int C>
var	sum (const Eigen::Matrix< var, R, C > &m)
	Returns the sum of the coefficients of the specified matrix, column vector or row vector. More...
template<int R, int C>
Eigen::Matrix< var,-1,-1 >	tcrossprod (const Eigen::Matrix< var, R, C > &M)
	Returns the result of post-multiplying a matrix by its own transpose. More...
matrix_v	to_var (const matrix_d &m)
	Converts argument to an automatic differentiation variable. More...
matrix_v &	to_var (matrix_v &m)
	Specialization of to_var for non-const matrices of vars. More...
const matrix_v &	to_var (const matrix_v &m)
	Specialization of to_var for const matrices of vars. More...
vector_v	to_var (const vector_d &v)
	Converts argument to an automatic differentiation variable. More...
const vector_v &	to_var (const vector_v &v)
	Specialization of to_var for const column vector of vars. More...
vector_v &	to_var (vector_v &v)
	Specialization of to_var for non-const column vector of vars. More...
row_vector_v	to_var (const row_vector_d &rv)
	Converts argument to an automatic differentiation variable. More...
const row_vector_v &	to_var (const row_vector_v &rv)
	Specialization of to_var for const row vector of vars. More...
row_vector_v &	to_var (row_vector_v &rv)
	Specialization of to_var for non-const row vector of vars. More...
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename T3 , int R3, int C3, require_any_var_t< T1, T2, T3 > ...>
var	trace_gen_inv_quad_form_ldlt (const Eigen::Matrix< T1, R1, C1 > &D, const LDLT_factor< T2, R2, C2 > &A, const Eigen::Matrix< T3, R3, C3 > &B)
	Compute the trace of an inverse quadratic form. More...
template<typename Td , int Rd, int Cd, typename Ta , int Ra, int Ca, typename Tb , int Rb, int Cb, typename = require_any_var_t<Td, Ta, Tb>>
var	trace_gen_quad_form (const Eigen::Matrix< Td, Rd, Cd > &D, const Eigen::Matrix< Ta, Ra, Ca > &A, const Eigen::Matrix< Tb, Rb, Cb > &B)
template<typename T2 , int R2, int C2, typename T3 , int R3, int C3, typename = require_any_var_t<T2, T3>>
return_type_t< T2, T3 >	trace_inv_quad_form_ldlt (const LDLT_factor< T2, R2, C2 > &A, const Eigen::Matrix< T3, R3, C3 > &B)
	Compute the trace of an inverse quadratic form. More...
template<typename Ta , int Ra, int Ca, typename Tb , int Rb, int Cb, typename = require_any_var_t<Ta, Tb>>
return_type_t< Ta, Tb >	trace_quad_form (const Eigen::Matrix< Ta, Ra, Ca > &A, const Eigen::Matrix< Tb, Rb, Cb > &B)
template<int R, int C>
Eigen::Matrix< var, R, C >	unit_vector_constrain (const Eigen::Matrix< var, R, C > &y)
	Return the unit length vector corresponding to the free vector y. More...
template<int R, int C>
Eigen::Matrix< var, R, C >	unit_vector_constrain (const Eigen::Matrix< var, R, C > &y, var &lp)
	Return the unit length vector corresponding to the free vector y. More...
var	variance (const std::vector< var > &v)
	Return the sample variance of the specified standard vector. More...
template<int R, int C>
var	variance (const Eigen::Matrix< var, R, C > &m)
template<typename F , typename... Targs>
auto	adj_jac_apply (const Targs &...args)
	Return the result of applying the function defined by a nullary construction of F to the specified input argument. More...
template<typename F , typename T1 , typename T2 , typename T_u , typename T_f >
Eigen::Matrix< T2,-1, 1 >	algebra_solver_fp (const F &f, const Eigen::Matrix< T1,-1, 1 > &x, const Eigen::Matrix< T2,-1, 1 > &y, const std::vector< double > &x_r, const std::vector< int > &x_i, const std::vector< T_u > &u_scale, const std::vector< T_f > &f_scale, std::ostream *msgs=nullptr, double f_tol=1e-8, int max_num_steps=200)
	Return a fixed pointer to the specified system of algebraic equations of form. More...
template<typename F , typename T >
Eigen::VectorXd	algebra_solver_newton (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, const Eigen::VectorXd &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *msgs=nullptr, double scaling_step_size=1e-3, double function_tolerance=1e-6, long int max_num_steps=200)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T1 , typename T2 >
Eigen::Matrix< T2, Eigen::Dynamic, 1 >	algebra_solver_newton (const F &f, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *msgs=nullptr, double scaling_step_size=1e-3, double function_tolerance=1e-6, long int max_num_steps=200)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T >
Eigen::VectorXd	algebra_solver_powell (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, const Eigen::VectorXd &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-10, double function_tolerance=1e-6, long int max_num_steps=1e+3)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T1 , typename T2 >
Eigen::Matrix< T2, Eigen::Dynamic, 1 >	algebra_solver_powell (const F &f, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-10, double function_tolerance=1e-6, long int max_num_steps=1e+3)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename F , typename T1 , typename T2 >
Eigen::Matrix< T2, Eigen::Dynamic, 1 >	algebra_solver (const F &f, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-10, double function_tolerance=1e-6, long int max_num_steps=1e+3)
	Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system. More...
template<typename T1 , typename T2 >
void	algebra_solver_check (const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > y, const std::vector< double > &dat, const std::vector< int > &dat_int, double function_tolerance, long int max_num_steps)
void	cvodes_silent_err_handler (int error_code, const char *module, const char *function, char *msg, void *eh_data)
void	cvodes_set_options (void *cvodes_mem, double rel_tol, double abs_tol, long int max_num_steps)
template<typename F >
void	gradient (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad_fx)
	Calculate the value and the gradient of the specified function at the specified argument. More...
template<typename F , typename Tpar >
std::vector< std::vector< Tpar > >	integrate_dae (const F &f, const std::vector< double > &yy0, const std::vector< double > &yp0, double t0, const std::vector< double > &ts, const std::vector< Tpar > &theta, const std::vector< double > &x_r, const std::vector< int > &x_i, const double rtol, const double atol, const int64_t max_num_steps=idas_integrator::IDAS_MAX_STEPS, std::ostream *msgs=nullptr)
	Return the solutions for a semi-explicit DAE system with residual specified by functor F, given the specified consistent initial state yy0 and yp0. More...
template<typename F , typename T_initial , typename T_param , typename T_t0 , typename T_ts >
std::vector< std::vector< typename stan::return_type< T_initial, T_param, T_t0, T_ts >::type > >	integrate_ode_adams (const F &f, const std::vector< T_initial > &y0, const T_t0 &t0, const std::vector< T_ts > &ts, const std::vector< T_param > &theta, const std::vector< double > &x, const std::vector< int > &x_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-10, double absolute_tolerance=1e-10, long int max_num_steps=1e8)
template<typename F , typename T_initial , typename T_param , typename T_t0 , typename T_ts >
std::vector< std::vector< typename stan::return_type< T_initial, T_param, T_t0, T_ts >::type > >	integrate_ode_bdf (const F &f, const std::vector< T_initial > &y0, const T_t0 &t0, const std::vector< T_ts > &ts, const std::vector< T_param > &theta, const std::vector< double > &x, const std::vector< int > &x_int, std::ostream *msgs=nullptr, double relative_tolerance=1e-10, double absolute_tolerance=1e-10, long int max_num_steps=1e8)
template<typename F >
void	jacobian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, Eigen::Matrix< double, Eigen::Dynamic, 1 > &fx, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &J)
template<typename F1 , typename F2 = kinsol_J_f>
Eigen::VectorXd	kinsol_solve (const F1 &f, const Eigen::VectorXd &x, const Eigen::VectorXd &y, const std::vector< double > &dat, const std::vector< int > &dat_int, std::ostream *msgs=nullptr, double scaling_step_tol=1e-3, double function_tolerance=1e-6, long int max_num_steps=200, bool custom_jacobian=1, const F2 &J_f=kinsol_J_f(), int steps_eval_jacobian=10, int global_line_search=KIN_LINESEARCH)
	Return the solution to the specified algebraic system, given an initial guess. More...
var	abs (const var &a)
	Return the absolute value of the variable (std). More...
var	acos (const var &a)
	Return the principal value of the arc cosine of a variable, in radians (cmath). More...
var	acosh (const var &a)
	The inverse hyperbolic cosine function for variables (C99). More...
int	as_bool (const var &v)
	Return 1 if the argument is unequal to zero and 0 otherwise. More...
var	asin (const var &a)
	Return the principal value of the arc sine, in radians, of the specified variable (cmath). More...
var	asinh (const var &a)
	The inverse hyperbolic sine function for variables (C99). More...
var	atan (const var &a)
	Return the principal value of the arc tangent, in radians, of the specified variable (cmath). More...
var	atan2 (const var &a, const var &b)
	Return the principal value of the arc tangent, in radians, of the first variable divided by the second (cmath). More...
var	atan2 (const var &a, double b)
	Return the principal value of the arc tangent, in radians, of the first variable divided by the second scalar (cmath). More...
var	atan2 (double a, const var &b)
	Return the principal value of the arc tangent, in radians, of the first scalar divided by the second variable (cmath). More...
var	atanh (const var &a)
	The inverse hyperbolic tangent function for variables (C99). More...
var	bessel_first_kind (int v, const var &a)
var	bessel_second_kind (int v, const var &a)
var	beta (const var &a, const var &b)
var	beta (const var &a, double b)
var	beta (double a, const var &b)
var	binary_log_loss (int y, const var &y_hat)
	The log loss function for variables (stan). More...
double	calculate_chain (double x, double val)
var	cbrt (const var &a)
	Returns the cube root of the specified variable (C99). More...
var	ceil (const var &a)
	Return the ceiling of the specified variable (cmath). More...
var	cos (const var &a)
	Return the cosine of a radian-scaled variable (cmath). More...
var	cosh (const var &a)
	Return the hyperbolic cosine of the specified variable (cmath). More...
var	digamma (const var &a)
var	erf (const var &a)
	The error function for variables (C99). More...
var	erfc (const var &a)
	The complementary error function for variables (C99). More...
var	exp (const var &a)
	Return the exponentiation of the specified variable (cmath). More...
var	exp2 (const var &a)
	Exponentiation base 2 function for variables (C99). More...
var	expm1 (const var &a)
	The exponentiation of the specified variable minus 1 (C99). More...
var	fabs (const var &a)
	Return the absolute value of the variable (cmath). More...
var	falling_factorial (const var &a, int b)
var	fdim (const var &a, const var &b)
	Return the positive difference between the first variable's the value and the second's (C99, C++11). More...
var	fdim (double a, const var &b)
	Return the positive difference between the first value and the value of the second variable (C99, C++11). More...
var	fdim (const var &a, double b)
	Return the positive difference between the first variable's value and the second value (C99, C++11). More...
var	floor (const var &a)
	Return the floor of the specified variable (cmath). More...
var	fma (const var &a, const var &b, const var &c)
	The fused multiply-add function for three variables (C99). More...
template<typename Tc , typename = require_arithmetic_t<Tc>>
var	fma (const var &a, const var &b, Tc &&c)
	The fused multiply-add function for two variables and a value (C99). More...
template<typename Ta , typename Tb , typename Tc , typename = require_arithmetic_t<Tb>, typename = require_all_var_t<Ta, Tc>>
var	fma (Ta &&a, Tb &&b, Tc &&c)
	The fused multiply-add function for a variable, value, and variable (C99). More...
template<typename Tb , typename Tc , typename = require_all_arithmetic_t<Tb, Tc>>
var	fma (const var &a, Tb &&b, Tc &&c)
	The fused multiply-add function for a variable and two values (C99). More...
template<typename Ta , typename Tc , typename = require_all_arithmetic_t<Ta, Tc>>
var	fma (Ta &&a, const var &b, Tc &&c)
	The fused multiply-add function for a value, variable, and value (C99). More...
template<typename Ta , typename Tb , typename = require_all_arithmetic_t<Ta, Tb>>
var	fma (Ta &&a, Tb &&b, const var &c)
	The fused multiply-add function for two values and a variable, and value (C99). More...
template<typename Ta , typename = require_arithmetic_t<Ta>>
var	fma (Ta &&a, const var &b, const var &c)
	The fused multiply-add function for a value and two variables (C99). More...
var	fmax (const var &a, const var &b)
	Returns the maximum of the two variable arguments (C99). More...
var	fmax (const var &a, double b)
	Returns the maximum of the variable and scalar, promoting the scalar to a variable if it is larger (C99). More...
var	fmax (double a, const var &b)
	Returns the maximum of a scalar and variable, promoting the scalar to a variable if it is larger (C99). More...
var	fmin (const var &a, const var &b)
	Returns the minimum of the two variable arguments (C99). More...
var	fmin (const var &a, double b)
	Returns the minimum of the variable and scalar, promoting the scalar to a variable if it is larger (C99). More...
var	fmin (double a, const var &b)
	Returns the minimum of a scalar and variable, promoting the scalar to a variable if it is larger (C99). More...
var	fmod (const var &a, const var &b)
	Return the floating point remainder after dividing the first variable by the second (cmath). More...
var	fmod (const var &a, double b)
	Return the floating point remainder after dividing the the first variable by the second scalar (cmath). More...
var	fmod (double a, const var &b)
	Return the floating point remainder after dividing the first scalar by the second variable (cmath). More...
var	gamma_p (const var &a, const var &b)
var	gamma_p (const var &a, double b)
var	gamma_p (double a, const var &b)
var	gamma_q (const var &a, const var &b)
var	gamma_q (const var &a, double b)
var	gamma_q (double a, const var &b)
void	grad_inc_beta (var &g1, var &g2, const var &a, const var &b, const var &z)
	Gradient of the incomplete beta function beta(a, b, z) with respect to the first two arguments. More...
var	hypot (const var &a, const var &b)
	Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...
var	hypot (const var &a, double b)
	Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...
var	hypot (double a, const var &b)
	Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...
var	ibeta (const var &a, const var &b, const var &x)
	The normalized incomplete beta function of a, b, and x. More...
var	if_else (bool c, const var &y_true, const var &y_false)
	If the specified condition is true, return the first variable, otherwise return the second variable. More...
var	if_else (bool c, double y_true, const var &y_false)
	If the specified condition is true, return a new variable constructed from the first scalar, otherwise return the second variable. More...
var	if_else (bool c, const var &y_true, double y_false)
	If the specified condition is true, return the first variable, otherwise return a new variable constructed from the second scalar. More...
var	inc_beta (const var &a, const var &b, const var &c)
var	inv (const var &a)
	\[ \mbox{inv}(x) = \begin{cases} \frac{1}{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \] More...
var	inv_cloglog (const var &a)
	Return the inverse complementary log-log function applied specified variable (stan). More...
var	inv_logit (const var &a)
	The inverse logit function for variables (stan). More...
var	inv_Phi (const var &p)
	The inverse of unit normal cumulative density function. More...
var	inv_sqrt (const var &a)
	\[ \mbox{inv\_sqrt}(x) = \begin{cases} \frac{1}{\sqrt{x}} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \] More...
var	inv_square (const var &a)
	\[ \mbox{inv\_square}(x) = \begin{cases} \frac{1}{x^2} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \] More...
int	is_inf (const var &v)
	Returns 1 if the input's value is infinite and 0 otherwise. More...
bool	is_uninitialized (var x)
	Returns true if the specified variable is uninitialized. More...
var	lbeta (const var &a, const var &b)
var	lbeta (const var &a, double b)
var	lbeta (double a, const var &b)
var	ldexp (const var &a, int b)
var	lgamma (const var &a)
	The log gamma function for variables (C99). More...
var	lmgamma (int a, const var &b)
var	log (const var &a)
	Return the natural log of the specified variable (cmath). More...
var	log10 (const var &a)
	Return the base 10 log of the specified variable (cmath). More...
var	log1m (const var &a)
	The log (1 - x) function for variables. More...
var	log1m_exp (const var &x)
	Return the log of 1 minus the exponential of the specified variable. More...
var	log1m_inv_logit (const var &u)
	Return the natural logarithm of one minus the inverse logit of the specified argument. More...
var	log1p (const var &a)
	The log (1 + x) function for variables (C99). More...
var	log1p_exp (const var &a)
	Return the log of 1 plus the exponential of the specified variable. More...
var	log2 (const var &a)
	Returns the base 2 logarithm of the specified variable (C99). More...
var	log_diff_exp (const var &a, const var &b)
	Returns the log difference of the exponentiated arguments. More...
var	log_diff_exp (const var &a, double b)
	Returns the log difference of the exponentiated arguments. More...
var	log_diff_exp (double a, const var &b)
	Returns the log difference of the exponentiated arguments. More...
var	log_falling_factorial (const var &a, double b)
var	log_falling_factorial (const var &a, const var &b)
var	log_falling_factorial (double a, const var &b)
var	log_inv_logit (const var &u)
	Return the natural logarithm of the inverse logit of the specified argument. More...
var	log_inv_logit_diff (const var &a, double b)
var	log_inv_logit_diff (const var &a, const var &b)
var	log_inv_logit_diff (double a, const var &b)
void	log_mix_partial_helper (double theta_val, double lambda1_val, double lambda2_val, double &one_m_exp_lam2_m_lam1, double &one_m_t_prod_exp_lam2_m_lam1, double &one_d_t_plus_one_m_t_prod_exp_lam2_m_lam1)
template<typename T_theta , typename T_lambda1 , typename T_lambda2 >
return_type_t< T_theta, T_lambda1, T_lambda2 >	log_mix (const T_theta &theta, const T_lambda1 &lambda1, const T_lambda2 &lambda2)
	Return the log mixture density with specified mixing proportion and log densities and its derivative at each. More...
var	log_rising_factorial (const var &a, double b)
var	log_rising_factorial (const var &a, const var &b)
var	log_rising_factorial (double a, const var &b)
var	log_sum_exp (const var &a, const var &b)
	Returns the log sum of exponentials. More...
var	log_sum_exp (const var &a, double b)
	Returns the log sum of exponentials. More...
var	log_sum_exp (double a, const var &b)
	Returns the log sum of exponentials. More...
var	logit (const var &u)
	Return the log odds of the specified argument. More...
var	modified_bessel_first_kind (int v, const var &a)
var	modified_bessel_second_kind (int v, const var &a)
var	multiply_log (const var &a, const var &b)
	Return the value of a*log(b). More...
var	multiply_log (const var &a, double b)
	Return the value of a*log(b). More...
var	multiply_log (double a, const var &b)
	Return the value of a*log(b). More...
var	owens_t (const var &h, const var &a)
	The Owen's T function of h and a. More...
var	owens_t (const var &h, double a)
	The Owen's T function of h and a. More...
var	owens_t (double h, const var &a)
	The Owen's T function of h and a. More...
var	Phi (const var &a)
	The unit normal cumulative density function for variables (stan). More...
var	Phi_approx (const var &a)
	Approximation of the unit normal CDF for variables (stan). More...
var	pow (const var &base, const var &exponent)
	Return the base raised to the power of the exponent (cmath). More...
var	pow (const var &base, double exponent)
	Return the base variable raised to the power of the exponent scalar (cmath). More...
var	pow (double base, const var &exponent)
	Return the base scalar raised to the power of the exponent variable (cmath). More...
double	primitive_value (const var &v)
	Return the primitive double value for the specified auto-diff variable. More...
var	rising_factorial (const var &a, int b)
var	round (const var &a)
	Returns the rounded form of the specified variable (C99). More...
var	sin (const var &a)
	Return the sine of a radian-scaled variable (cmath). More...
var	sinh (const var &a)
	Return the hyperbolic sine of the specified variable (cmath). More...
var	sqrt (const var &a)
	Return the square root of the specified variable (cmath). More...
var	square (const var &x)
	Return the square of the input variable. More...
var	squared_distance (const var &a, const var &b)
	Returns the log sum of exponentials. More...
var	squared_distance (const var &a, double b)
	Returns the log sum of exponentials. More...
var	squared_distance (double a, const var &b)
	Returns the log sum of exponentials. More...
var	step (const var &a)
	Return the step, or heaviside, function applied to the specified variable (stan). More...
var	tan (const var &a)
	Return the tangent of a radian-scaled variable (cmath). More...
var	tanh (const var &a)
	Return the hyperbolic tangent of the specified variable (cmath). More...
var	tgamma (const var &a)
	Return the Gamma function applied to the specified variable (C99). More...
var	to_var (double x)
	Converts argument to an automatic differentiation variable. More...
var &	to_var (var &x)
	Specialization of to_var for non-const var input. More...
const var &	to_var (const var &x)
	Specialization of to_var for const var input. More...
var	trigamma (const var &u)
	Return the value of the trigamma function at the specified argument (i.e., the second derivative of the log Gamma function at the specified argument). More...
var	trunc (const var &a)
	Returns the truncatation of the specified variable (C99). More...
double	value_of (const var &v)
	Return the value of the specified variable. More...
double	value_of_rec (const var &v)
	Return the value of the specified variable. More...

Variables
static opencl_context	opencl_context
const double	CONSTRAINT_TOLERANCE = 1E-8
	The tolerance for checking arithmetic bounds In rank and in simplexes. More...
const double	E = boost::math::constants::e<double>()
	The base of the natural logarithm, \( e \). More...
const double	SQRT_TWO = std::sqrt(2.0)
	The value of the square root of 2, \( \sqrt{2} \). More...
const double	INV_SQRT_TWO = inv(SQRT_TWO)
	The value of 1 over the square root of 2, \( 1 / \sqrt{2} \). More...
const double	LOG_TWO = std::log(2.0)
	The natural logarithm of 2, \( \log 2 \). More...
const double	LOG_TEN = std::log(10.0)
	The natural logarithm of 10, \( \log 10 \). More...
const double	INFTY = std::numeric_limits<double>::infinity()
	Positive infinity. More...
const double	NEGATIVE_INFTY = -INFTY
	Negative infinity. More...
const double	NOT_A_NUMBER = std::numeric_limits<double>::quiet_NaN()
	(Quiet) not-a-number value. More...
const double	EPSILON = std::numeric_limits<double>::epsilon()
	Smallest positive value. More...
const double	NEGATIVE_EPSILON = -EPSILON
	Largest negative value (i.e., smallest absolute value). More...
const double	POISSON_MAX_RATE = std::pow(2.0, 30)
	Largest rate parameter allowed in Poisson RNG. More...
const double	SQRT_PI = std::sqrt(pi())
const double	INV_SQRT_PI = inv(SQRT_PI)
const double	TWO_OVER_SQRT_PI = 2.0 / SQRT_PI
const double	NEG_TWO_OVER_SQRT_PI = -TWO_OVER_SQRT_PI
const double	SQRT_TWO_OVER_SQRT_PI = SQRT_TWO / SQRT_PI
const double	TWO_PI = 2.0 * pi()
const double	SQRT_TWO_PI = std::sqrt(TWO_PI)
const double	INV_SQRT_TWO_PI = inv(SQRT_TWO_PI)
const double	LOG_PI = std::log(pi())
const double	LOG_PI_OVER_FOUR = LOG_PI / 4.0
const double	LOG_SQRT_PI = std::log(SQRT_PI)
const double	LOG_ZERO = std::log(0.0)
const double	LOG_HALF = std::log(0.5)
const double	NEG_LOG_TWO = -LOG_TWO
const double	NEG_LOG_SQRT_TWO_PI = -std::log(SQRT_TWO_PI)
const double	NEG_LOG_PI = -LOG_PI
const double	NEG_LOG_SQRT_PI = -LOG_SQRT_PI
const double	NEG_LOG_TWO_OVER_TWO = -LOG_TWO / 2.0
const double	LOG_TWO_PI = LOG_TWO + LOG_PI
const double	NEG_LOG_TWO_PI = -LOG_TWO_PI
const double	LOG_EPSILON = std::log(EPSILON)
const std::string	MAJOR_VERSION = STAN_STRING(STAN_MATH_MAJOR)
	Major version number for Stan math library. More...
const std::string	MINOR_VERSION = STAN_STRING(STAN_MATH_MINOR)
	Minor version number for Stan math library. More...
const std::string	PATCH_VERSION = STAN_STRING(STAN_MATH_PATCH)
	Patch version for Stan math library. More...

Typedef Documentation

template<typename T >

using stan::math::as_operation_cl_t = typedef std::conditional_t< std::is_lvalue_reference<T>::value, decltype(as_operation_cl(std::declval<T>())), std::remove_reference_t<decltype(as_operation_cl(std::declval<T>()))>>

Type that results when converting any valid kernel generator expression into operation.

If a function accepts a forwarding reference T&& a, the result of as_operation_cl(a) should be stored in a variable of type as_operation_cl_t<T>. If the return value of as_operation_cl() would be a rvalue reference, the reference is removed, so that a variable of this type actually stores the value.

Definition at line 66 of file as_operation_cl.hpp.

using stan::math::boost_policy_t = typedef boost::math::policies::policy< boost::math::policies::overflow_error< boost::math::policies::errno_on_error>, boost::math::policies::pole_error<boost::math::policies::errno_on_error>, boost::math::policies::promote_double<false> >

Boost policy that overrides the defaults to match the built-in C++ standard library functions.

The non-default behavior from Boost's built-ins are (1) overflow errors return error numbers on error. (2) pole errors return error numbers on error.

Definition at line 23 of file boost_policy.hpp.

using stan::math::ChainableStack = typedef AutodiffStackSingleton<vari, chainable_alloc>

Definition at line 12 of file chainablestack.hpp.

template<typename T >

typedef stan::math::index_type_t< Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >> stan::math::size_type

Type for sizes and indexes in an Eigen matrix with double e.

Definition at line 37 of file index_type.hpp.

template<typename T >

using stan::math::matrix_cl_fp = typedef matrix_cl<T, require_floating_point_t<T>>

Definition at line 604 of file matrix_cl.hpp.

template<typename T >

using stan::math::matrix_cl_prim = typedef matrix_cl<T, require_arithmetic_t<T>>

Definition at line 601 of file matrix_cl.hpp.

using stan::math::matrix_d = typedef Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic>

Type for matrix of double values.

Definition at line 19 of file typedefs.hpp.

using stan::math::matrix_fd = typedef Eigen::Matrix<fvar<double>, Eigen::Dynamic, Eigen::Dynamic>

Definition at line 13 of file typedefs.hpp.

using stan::math::matrix_ffd = typedef Eigen::Matrix<fvar<fvar<double> >, Eigen::Dynamic, Eigen::Dynamic>

Definition at line 16 of file typedefs.hpp.

using stan::math::matrix_ffv = typedef Eigen::Matrix<fvar<fvar<var>>, Eigen::Dynamic, Eigen::Dynamic>

Definition at line 15 of file typedefs.hpp.

using stan::math::matrix_fv = typedef Eigen::Matrix<fvar<var>, Eigen::Dynamic, Eigen::Dynamic>

Definition at line 12 of file typedefs.hpp.

using stan::math::matrix_v = typedef Eigen::Matrix<var, Eigen::Dynamic, Eigen::Dynamic>

The type of a matrix holding var values.

Definition at line 17 of file typedefs.hpp.

using stan::math::matrix_vi = typedef Eigen::Matrix<vari*, Eigen::Dynamic, Eigen::Dynamic>

The type of a matrix holding vari* values.

Definition at line 35 of file typedefs.hpp.

template<typename T , typename S >

using stan::math::promote_scalar_t = typedef typename promote_scalar_type<T, S>::type

Definition at line 24 of file promote_scalar_type.hpp.

template<typename... Types>

using stan::math::require_all_valid_expressions_and_none_scalar_t = typedef require_all_t<is_valid_expression_and_not_scalar<Types>...>

Enables a template if all given types are non-scalar types that are a valid kernel generator expressions.

Definition at line 48 of file is_valid_expression.hpp.

template<typename... Types>

using stan::math::require_all_valid_expressions_t = typedef require_all_t<is_valid_expression<Types>...>

Enables a template if all given types are are a valid kernel generator expressions.

Definition at line 56 of file is_valid_expression.hpp.

using stan::math::row_vector_d = typedef Eigen::Matrix<double, 1, Eigen::Dynamic>

Type for (row) vector of double values.

Definition at line 29 of file typedefs.hpp.

using stan::math::row_vector_fd = typedef Eigen::Matrix<fvar<double>, 1, Eigen::Dynamic>

Definition at line 22 of file typedefs.hpp.

using stan::math::row_vector_ffd = typedef Eigen::Matrix<fvar<fvar<double> >, 1, Eigen::Dynamic>

Definition at line 24 of file typedefs.hpp.

using stan::math::row_vector_ffv = typedef Eigen::Matrix<fvar<fvar<var>>, 1, Eigen::Dynamic>

Definition at line 23 of file typedefs.hpp.

using stan::math::row_vector_fv = typedef Eigen::Matrix<fvar<var>, 1, Eigen::Dynamic>

Definition at line 21 of file typedefs.hpp.

using stan::math::row_vector_v = typedef Eigen::Matrix<var, 1, Eigen::Dynamic>

The type of a row vector holding var values.

Definition at line 29 of file typedefs.hpp.

using stan::math::row_vector_vi = typedef Eigen::Matrix<vari*, 1, Eigen::Dynamic>

The type of a row vector holding vari* values.

Definition at line 47 of file typedefs.hpp.

typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >::Index stan::math::size_type

Type for sizes and indexes in an Eigen matrix with double e.

Definition at line 11 of file typedefs.hpp.

using stan::math::vector_d = typedef Eigen::Matrix<double, Eigen::Dynamic, 1>

Type for (column) vector of double values.

Definition at line 24 of file typedefs.hpp.

using stan::math::vector_fd = typedef Eigen::Matrix<fvar<double>, Eigen::Dynamic, 1>

Definition at line 18 of file typedefs.hpp.

using stan::math::vector_ffd = typedef Eigen::Matrix<fvar<fvar<double> >, Eigen::Dynamic, 1>

Definition at line 20 of file typedefs.hpp.

using stan::math::vector_ffv = typedef Eigen::Matrix<fvar<fvar<var>>, Eigen::Dynamic, 1>

Definition at line 19 of file typedefs.hpp.

using stan::math::vector_fv = typedef Eigen::Matrix<fvar<var>, Eigen::Dynamic, 1>

Definition at line 17 of file typedefs.hpp.

using stan::math::vector_v = typedef Eigen::Matrix<var, Eigen::Dynamic, 1>

The type of a (column) vector holding var values.

Definition at line 23 of file typedefs.hpp.

using stan::math::vector_vi = typedef Eigen::Matrix<vari*, Eigen::Dynamic, 1>

The type of a (column) vector holding vari* values.

Definition at line 41 of file typedefs.hpp.

Enumeration Type Documentation

enum stan::math::matrix_cl_view

strong

Enumerator
Diagonal
Lower
Upper
Entire

Definition at line 11 of file matrix_cl_view.hpp.

enum stan::math::TriangularMapCL

strong

Enumerator
UpperToLower
LowerToUpper

Definition at line 96 of file matrix_cl_view.hpp.

Function Documentation

template<typename T >

fvar<T> stan::math::abs ( const fvar< T > &  x )

inline

Definition at line 14 of file abs.hpp.

double stan::math::abs ( double  x )

inline

Return floating-point absolute value.

Delegates to fabs(double) rather than std::abs(int).

Parameters

x

scalar

Returns

absolute value of scalar

Definition at line 19 of file abs.hpp.

var stan::math::abs ( const var &  a )

inline

Return the absolute value of the variable (std).

Delegates to fabs() (see for doc).

\[ \mbox{abs}(x) = \begin{cases} |x| & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{abs}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } x < 0 \\ 0 & \mbox{if } x = 0 \\ 1 & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable input.

Returns

Absolute value of variable.

Definition at line 36 of file abs.hpp.

template<typename T >

fvar<T> stan::math::acos ( const fvar< T > &  x )

inline

Definition at line 13 of file acos.hpp.

template<typename T >

apply_scalar_unary<acos_fun, T>::return_t stan::math::acos ( const T &  x )

inline

Vectorized version of acos().

Template Parameters

T	type of container

Parameters

x

container

Returns

Arc cosine of each variable in the container, in radians.

Definition at line 33 of file acos.hpp.

var stan::math::acos ( const var &  a )

inline

Return the principal value of the arc cosine of a variable, in radians (cmath).

The derivative is defined by

\(\frac{d}{dx} \arccos x = \frac{-1}{\sqrt{1 - x^2}}\).

\[ \mbox{acos}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \arccos(x) & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{acos}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \arccos(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x < -1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arccos(x)}{\partial x} = -\frac{1}{\sqrt{1-x^2}} \]

Parameters

a	Variable in range [-1, 1].

Returns

Arc cosine of variable, in radians.

Definition at line 57 of file acos.hpp.

template<typename T >

fvar<T> stan::math::acosh ( const fvar< T > &  x )

inline

Definition at line 14 of file acosh.hpp.

double stan::math::acosh ( double  x )

inline

Return the inverse hyperbolic cosine of the specified value.

Returns nan for nan argument.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic cosine of the argument.

Exceptions

std::domain_error

If argument is less than 1.

Definition at line 21 of file acosh.hpp.

template<typename T >

apply_scalar_unary<acosh_fun, T>::return_t stan::math::acosh ( const T &  x )

inline

Return the elementwise application of acosh() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise acosh of members of container.

Definition at line 38 of file acosh.hpp.

double stan::math::acosh ( int  x )

inline

Integer version of acosh.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic cosine of the argument.

Exceptions

std::domain_error

If argument is less than 1.

Definition at line 41 of file acosh.hpp.

var stan::math::acosh ( const var &  a )

inline

The inverse hyperbolic cosine function for variables (C99).

For non-variable function, see acosh().

The derivative is defined by

\(\frac{d}{dx} \mbox{acosh}(x) = \frac{x}{x^2 - 1}\).

\[ \mbox{acosh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \cosh^{-1}(x) & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{acosh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \frac{\partial\, \cosh^{-1}(x)}{\partial x} & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right) \]

\[ \frac{\partial \, \cosh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2-1}} \]

Parameters

a	The variable.

Returns

Inverse hyperbolic cosine of the variable.

Definition at line 61 of file acosh.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::add ( const Eigen::Matrix< T1, R, C > &  m1, const Eigen::Matrix< T2, R, C > &  m2  )

inline

Return the sum of the specified matrices.

The two matrices must have the same dimensions.

Template Parameters

T1	type of elements in the first matrix
T2	type of elements in the second matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m1	First matrix.
m2	Second matrix.

Returns

Sum of the matrices.

Exceptions

std::invalid_argument

if m1 and m2 do not have the same dimensions.

Definition at line 26 of file add.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::add ( const Eigen::Matrix< T1, R, C > &  m, const T2 &  c  )

inline

Return the sum of the specified matrix and specified scalar.

Template Parameters

T1	type of elements in the matrix
T2	type of scalar
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	Matrix.
c	Scalar.

Returns

The matrix plus the scalar.

Definition at line 44 of file add.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::add ( const T1 &  c, const Eigen::Matrix< T2, R, C > &  m  )

inline

Return the sum of the specified scalar and specified matrix.

Template Parameters

T1	type of scalar
T2	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

c	Scalar.
m	Matrix.

Returns

The scalar plus the matrix.

Definition at line 61 of file add.hpp.

template<typename T_m , typename T_a >

Eigen::Matrix<return_type_t<T_m, T_a>, Eigen::Dynamic, Eigen::Dynamic> stan::math::add_diag ( const Eigen::Matrix< T_m, Eigen::Dynamic, Eigen::Dynamic > &  mat, const T_a &  to_add  )

inline

Returns a Matrix with values added along the main diagonal.

Template Parameters

T_m	type of element in Eigen::Matrix
T_a	Type of element to add along the diagonal

Parameters

mat	a matrix
to_add	value to add along the diagonal

Returns

a matrix with to_add added along main diagonal

Definition at line 25 of file add_diag.hpp.

template<typename T_m , typename T_a , int R, int C>

Eigen::Matrix<return_type_t<T_m, T_a>, Eigen::Dynamic, Eigen::Dynamic> stan::math::add_diag ( const Eigen::Matrix< T_m, Eigen::Dynamic, Eigen::Dynamic > &  mat, const Eigen::Matrix< T_a, R, C > &  to_add  )

inline

Returns a Matrix with values added along the main diagonal.

Template Parameters

T_m	type of element in Eigen::Matrix
T_a	Type of element to add along the diagonal
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

mat	a matrix
to_add	Sequence of values to add along the diagonal

Returns

a matrix with to_add added along main diagonal

Exceptions

invalid_argument

if to_add is vector-like but does not have the same number of elements as the main diagonal of mat

Definition at line 50 of file add_diag.hpp.

template<typename F , typename... Targs>

auto stan::math::adj_jac_apply

(

const Targs &...

args

)

Return the result of applying the function defined by a nullary construction of F to the specified input argument.

adj_jac_apply makes it possible to write efficient reverse-mode autodiff code without ever touching Stan's autodiff internals

Mathematically, to use a function in reverse mode autodiff, you need to be able to evaluate the function (y = f(x)) and multiply the Jacobian of that function (df(x)/dx) by a vector.

As an example, pretend there exists some large, complicated function, L(x1, x2), which contains our smaller function f(x1, x2). The goal of autodiff is to compute the partials dL/dx1 and dL/dx2. If we break the large function into pieces:

y = f(x1, x2) L = g(y)

If we were given dL/dy we could compute dL/dx1 by the product dL/dy * dy/dx1 or dL/dx2 by the product dL/dy * dy/dx2

Because y = f(x1, x2), dy/dx1 is just df(x1, x2)/dx1, the Jacobian of the function we're trying to define with x2 held fixed. A similar thing happens for dy/dx2. In vector form,

dL/dx1 = (dL/dy)' * df(x1, x2)/dx1 and dL/dx2 = (dL/dy)' * df(x1, x2)/dx2

So implementing f(x1, x2) and the products above are all that is required mathematically to implement reverse-mode autodiff for a function.

adj_jac_apply takes as a template argument a functor F that supplies the non-static member functions (leaving exact template arguments off):

(required) Eigen::VectorXd operator()(const std::array<bool, size>& needs_adj, const T1& x1..., const T2& x2, ...)

where there can be any number of input arguments x1, x2, ... and T1, T2, ... can be either doubles or any Eigen::Matrix type with double scalar values. needs_adj is an array of size equal to the number of input arguments indicating whether or not the adjoint^T Jacobian product must be computed for each input argument. This argument is passed to operator() so that any unnecessary preparatory calculations for multiply_adjoint_jacobian can be avoided if possible.

(required) std::tuple<T1, T2, ...> multiply_adjoint_jacobian(const std::array<bool, size>& needs_adj, const Eigen::VectorXd& adj)

where T1, T2, etc. are the same types as in operator(), needs_adj is the same as in operator(), and adj is the vector dL/dy.

operator() is responsible for computing f(x) and multiply_adjoint_jacobian is responsible for computing the necessary adjoint transpose Jacobian products (which frequently does not require the calculation of the full Jacobian).

operator() will be called before multiply_adjoint_jacobian is called, and is only called once in the lifetime of the functor multiply_adjoint_jacobian is called after operator() and may be called multiple times for any single functor

The functor supplied to adj_jac_apply must be careful to allocate any variables it defines in the autodiff arena because its destructor will never be called and memory will leak if allocated anywhere else.

Targs (the input argument types) can be any mix of doubles, vars, ints, std::vectors with double, var, or int scalar components, or Eigen::Matrix s of any shape with var or double scalar components

Template Parameters

F	functor to be connected to the autodiff stack
Targs	types of arguments to pass to functor

Parameters

args	input to the functor

Returns

the result of the specified operation wrapped up in vars

Definition at line 617 of file adj_jac_apply.hpp.

template<typename F , typename T1 , typename T2 >

Eigen::Matrix<T2, Eigen::Dynamic, 1> stan::math::algebra_solver	(	const F &	f,
		const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *	msgs = nullptr,
		double	relative_tolerance = 1e-10,
		double	function_tolerance = 1e-6,
		long int	max_num_steps = 1e+3
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use Powell's dogleg solver.

The user can also specify the relative tolerance (xtol in Eigen's code), the function tolerance, and the maximum number of steps (maxfev in Eigen's code).

Signature to maintain backward compatibility, will be removed in the future.

Template Parameters

F	type of equation system function.
T1	Type of elements in x vector.
T2	Type of elements in y vector.

Parameters

[in]	f	Functor that evaluates the system of equations.
[in]	x	Vector of starting values (initial guess).
[in]	y	parameter vector for the equation system.
[in]	dat	continuous data vector for the equation system.
[in]	dat_int	integer data vector for the equation system.
[in,out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.

Returns

theta Vector of solutions to the system of equations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if relative_tolerance is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if solver exceeds max_num_steps.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if the norm of the solution exceeds the function tolerance.

Definition at line 305 of file algebra_solver_powell.hpp.

template<typename T1 , typename T2 >

void stan::math::algebra_solver_check	(	const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T2, Eigen::Dynamic, 1 >	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		double	function_tolerance,
		long int	max_num_steps
	)

Definition at line 158 of file algebra_system.hpp.

template<typename F , typename T1 , typename T2 , typename T_u , typename T_f >

Eigen::Matrix<T2, -1, 1> stan::math::algebra_solver_fp	(	const F &	f,
		const Eigen::Matrix< T1,-1, 1 > &	x,
		const Eigen::Matrix< T2,-1, 1 > &	y,
		const std::vector< double > &	x_r,
		const std::vector< int > &	x_i,
		const std::vector< T_u > &	u_scale,
		const std::vector< T_f > &	f_scale,
		std::ostream *	msgs = nullptr,
		double	f_tol = 1e-8,
		int	max_num_steps = 200
	)

Return a fixed pointer to the specified system of algebraic equations of form.

x = F(x; theta)

given an initial guess x, and parameters theta and data. Use the KINSOL solver from the SUNDIALS suite.

The user can also specify the scaling controls, the function tolerance, and the maximum number of steps.

Template Parameters

F	type of equation system function.
T	type of initial guess vector. The final soluton type doesn't depend on initial guess type, but we allow initial guess to be either data or param.
T_u	type of scaling vector for unknowns. We allow it to be var because scaling could be parameter dependent. Internally these params are converted to data because scaling is applied.
T_f	type of scaling vector for residual. We allow it to be var because scaling could be parameter dependent. Internally these params are converted to data because scaling is applied.

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in]	y	Parameter vector for the equation system. The function is overloaded to treat y as a vector of doubles or of a a template type T.
[in]	x_r	Continuous data vector for the equation system.
[in]	x_i	Integer data vector for the equation system.
[in,out]	msgs	The print stream for warning messages.
[in]	u_scale	diagonal scaling matrix elements Du such that Du*x has all components roughly the same magnitude when x is close to a solution. (ref. KINSOL user guide chap.2 sec. "Scaling")
[in]	f_scale	diagonal scaling matrix elements such that Df*(x-f(x)) has all components roughly the same magnitude when x is not too close to a solution. (ref. KINSOL user guide chap.2 sec. "Scaling")
[in]	f_tol	Function-norm stopping tolerance.
[in]	max_num_steps	maximum number of function evaluations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if scaled_step_size is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if solver exceeds max_num_steps.

Definition at line 364 of file algebra_solver_fp.hpp.

template<typename F , typename T >

Eigen::VectorXd stan::math::algebra_solver_newton	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		const Eigen::VectorXd &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *	msgs = nullptr,
		double	scaling_step_size = 1e-3,
		double	function_tolerance = 1e-6,
		long int	max_num_steps = 200
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use the KINSOL solver from the SUNDIALS suite.

The user can also specify the scaled step size, the function tolerance, and the maximum number of steps.

Template Parameters

F	type of equation system function.
T	type of initial guess vector.

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in]	y	Parameter vector for the equation system. The function is overloaded to treat y as a vector of doubles or of a a template type T.
[in]	dat	Continuous data vector for the equation system.
[in]	dat_int	Integer data vector for the equation system.
[in,out]	msgs	The print stream for warning messages.
[in]	scaling_step_size	Scaled-step stopping tolerance. If a Newton step is smaller than the scaling step tolerance, the code breaks, assuming the solver is no longer making significant progress (i.e. is stuck)
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if scaled_step_size is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if solver exceeds max_num_steps.

Definition at line 58 of file algebra_solver_newton.hpp.

template<typename F , typename T1 , typename T2 >

Eigen::Matrix<T2, Eigen::Dynamic, 1> stan::math::algebra_solver_newton	(	const F &	f,
		const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *	msgs = nullptr,
		double	scaling_step_size = 1e-3,
		double	function_tolerance = 1e-6,
		long int	max_num_steps = 200
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use the KINSOL solver from the SUNDIALS suite.

The user can also specify the scaled step size, the function tolerance, and the maximum number of steps.

Overload the previous definition to handle the case where y is a vector of parameters (var). The overload calls the algebraic solver defined above and builds a vari object on top, using the algebra_solver_vari class.

Template Parameters

F	type of equation system function.
T	type of initial guess vector.

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in]	y	Parameter vector for the equation system. The function is overloaded to treat y as a vector of doubles or of a a template type T.
[in]	dat	Continuous data vector for the equation system.
[in]	dat_int	Integer data vector for the equation system.
[in,out]	msgs	The print stream for warning messages.
[in]	scaling_step_size	Scaled-step stopping tolerance. If a Newton step is smaller than the scaling step tolerance, the code breaks, assuming the solver is no longer making significant progress (i.e. is stuck)
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.

Returns

theta Vector of solutions to the system of equations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if scaled_step_size is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if solver exceeds max_num_steps.

Definition at line 120 of file algebra_solver_newton.hpp.

template<typename F , typename T >

Eigen::VectorXd stan::math::algebra_solver_powell	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		const Eigen::VectorXd &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *	msgs = nullptr,
		double	relative_tolerance = 1e-10,
		double	function_tolerance = 1e-6,
		long int	max_num_steps = 1e+3
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use Powell's dogleg solver.

The user can also specify the relative tolerance (xtol in Eigen's code), the function tolerance, and the maximum number of steps (maxfev in Eigen's code).

Throw an exception if the norm of f(x), where f is the output of the algebraic system and x the proposed solution, is greater than the function tolerance. We here use the norm as a metric to measure how far we are from the origin (0).

Template Parameters

F	type of equation system function.
T	type of initial guess vector.

Parameters

[in]	f	Functor that evaluates the system of equations.
[in]	x	Vector of starting values.
[in]	y	parameter vector for the equation system. The function is overloaded to treat y as a vector of doubles or of a a template type T.
[in]	dat	continuous data vector for the equation system.
[in]	dat_int	integer data vector for the equation system.
[in,out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.

Returns

theta Vector of solutions to the system of equations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if relative_tolerance is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if solver exceeds max_num_steps.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if the norm of the solution exceeds the function tolerance.

Definition at line 127 of file algebra_solver_powell.hpp.

template<typename F , typename T1 , typename T2 >

Eigen::Matrix<T2, Eigen::Dynamic, 1> stan::math::algebra_solver_powell	(	const F &	f,
		const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *	msgs = nullptr,
		double	relative_tolerance = 1e-10,
		double	function_tolerance = 1e-6,
		long int	max_num_steps = 1e+3
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use Powell's dogleg solver.

The user can also specify the relative tolerance (xtol in Eigen's code), the function tolerance, and the maximum number of steps (maxfev in Eigen's code).

Overload the previous definition to handle the case where y is a vector of parameters (var). The overload calls the algebraic solver defined above and builds a vari object on top, using the algebra_solver_vari class.

Template Parameters

F	type of equation system function.
T1	Type of elements in x vector.
T2	Type of elements in y vector.

Parameters

[in]	f	Functor that evaluates the system of equations.
[in]	x	Vector of starting values (initial guess).
[in]	y	parameter vector for the equation system.
[in]	dat	continuous data vector for the equation system.
[in]	dat_int	integer data vector for the equation system.
[in,out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.

Returns

theta Vector of solutions to the system of equations.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements.
<code>std::invalid_argument</code>	if y has non-finite elements.
<code>std::invalid_argument</code>	if dat has non-finite elements.
<code>std::invalid_argument</code>	if dat_int has non-finite elements.
<code>std::invalid_argument</code>	if relative_tolerance is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if solver exceeds max_num_steps.
<code>boost::math::evaluation_error</code>	(which is a subclass of std::runtime_error) if the norm of the solution exceeds the function tolerance.

Definition at line 227 of file algebra_solver_powell.hpp.

template<typename T1 , typename T2 >

append_return_type<std::vector<T1>, std::vector<T2> >::type stan::math::append_array ( const std::vector< T1 > &  x, const std::vector< T2 > &  y  )

inline

Return the concatenation of two specified vectors in the order of the arguments.

The return type is computed with append_return_type

Template Parameters

T1	type of elements in first vector
T2	type of elements in second vector

Parameters

x	First vector
y	Second vector

Returns

A vector of x and y concatenated together (in that order)

Definition at line 29 of file append_array.hpp.

template<typename T1 >

std::vector<T1> stan::math::append_array ( const std::vector< T1 > &  x, const std::vector< T1 > &  y  )

inline

Return the concatenation of two specified vectors in the order of the arguments.

Template Parameters

T1	type of elements in both vectors

Parameters

x	First vector
y	Second vector

Returns

A vector of x and y concatenated together (in that order)

Definition at line 59 of file append_array.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T1, R1, C1 > &  A, const Eigen::Matrix< T2, R2, C2 > &  B  )

inline

Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix.

The inputs can be (matrix, matrix), (matrix, vector), (vector, matrix), or (vector, vector) and the output is always a matrix.

Template Parameters

T1	type of elements in the first matrix
T2	type of elements in the second matrix
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

A	First matrix.
B	Second matrix.

Returns

Result of appending the first matrix followed by the second matrix side by side.

Definition at line 38 of file append_col.hpp.

template<typename T1 , typename T2 , int C1, int C2>

Eigen::Matrix<return_type_t<T1, T2>, 1, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T1, 1, C1 > &  A, const Eigen::Matrix< T2, 1, C2 > &  B  )

inline

Return the result of concatenating the first row vector followed by the second row vector side by side, with the result being a row vector.

This function applies to (row_vector, row_vector) and returns a row_vector.

Template Parameters

T1	type of elements in the first row vector
T2	type of elements in the second row vector
C1	number of columns in the first row vector, can be Eigen::Dynamic
C2	number of columns in the second row vector, can be Eigen::Dynamic

Parameters

A	First vector.
B	Second vector

Returns

Result of appending the second row vector to the right of the first row vector.

Definition at line 83 of file append_col.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T, R1, C1 > &  A, const Eigen::Matrix< T, R2, C2 > &  B  )

inline

Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix.

This is an overloaded template function for the case when both matrices have the same type.

The inputs can be (matrix, matrix), (matrix, vector), (vector, matrix), or (vector, vector), and the output is always a matrix.

Template Parameters

T	type of elements in both matrices
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

A	First matrix.
B	Second matrix.

Returns

Result of appending the first matrix followed by the second matrix side by side.

Definition at line 126 of file append_col.hpp.

template<typename T , int C1, int C2>

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T, 1, C1 > &  A, const Eigen::Matrix< T, 1, C2 > &  B  )

inline

Return the result of concatenating the first row vector followed by the second row vector side by side, with the result being a row vector.

This function applies to (row_vector, row_vector) and returns a row_vector.

Template Parameters

T	type of elements in both vectors
C1	number of columns in the first row vector, can be Eigen::Dynamic
C2	number of columns in the second row vector, can be Eigen::Dynamic

Parameters

A	First vector.
B	Second vector

Returns

Result of appending the second row vector to the right of the first row vector.

Definition at line 156 of file append_col.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, 1, Eigen::Dynamic> stan::math::append_col ( const T1 &  A, const Eigen::Matrix< T2, R, C > &  B  )

inline

Return the result of stacking an scalar on top of the a row vector, with the result being a row vector.

This function applies to (scalar, row vector) and returns a row vector.

Template Parameters

T1	type of the scalar
T2	type of elements in the row vector
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

A	scalar.
B	row vector.

Returns

Result of stacking the scalar on top of the row vector.

Definition at line 183 of file append_col.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, 1, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T1, R, C > &  A, const T2 &  B  )

inline

Return the result of stacking a row vector on top of the an scalar, with the result being a row vector.

This function applies to (row vector, scalar) and returns a row vector.

Template Parameters

T1	type of elements in the row vector
T2	type of the scalar
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

A	row vector.
B	scalar.

Returns

Result of stacking the row vector on top of the scalar.

Definition at line 211 of file append_col.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, Eigen::Dynamic> stan::math::append_row ( const Eigen::Matrix< T1, R1, C1 > &  A, const Eigen::Matrix< T2, R2, C2 > &  B  )

inline

Return the result of stacking the rows of the first argument matrix on top of the second argument matrix.

The inputs can be (matrix, matrix), (matrix, row_vector), (row_vector, matrix), or (row_vector, row_vector), and the output is always a matrix.

Template Parameters

T1	type of elements in first matrix
T2	type of elements in second matrix
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

A	First matrix.
B	Second matrix.

Returns

Result of stacking first matrix on top of second.

Definition at line 36 of file append_row.hpp.

template<typename T1 , typename T2 , int R1, int R2>

Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, 1> stan::math::append_row ( const Eigen::Matrix< T1, R1, 1 > &  A, const Eigen::Matrix< T2, R2, 1 > &  B  )

inline

Return the result of stacking the first vector on top of the second vector, with the result being a vector.

This function applies to (vector, vector) and returns a vector.

Template Parameters

T1	type of elements in first vector
T2	type of elements in second vector
R1	number of rows in the first vector, can be Eigen::Dynamic
R2	number of rows in the second vector, can be Eigen::Dynamic

Parameters

A	First vector.
B	Second vector.

Returns

Result of stacking first vector on top of the second vector.

Definition at line 76 of file append_row.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::append_row ( const Eigen::Matrix< T, R1, C1 > &  A, const Eigen::Matrix< T, R2, C2 > &  B  )

inline

Return the result of stacking the rows of the first argument matrix on top of the second argument matrix.

This is an overload for the case when the scalar types of the two input matrix are the same.

The inputs can be (matrix, matrix), (matrix, row_vector), (row_vector, matrix), or (row_vector, row_vector), and the output is always a matrix.

Template Parameters

T	type of elements in both matrices
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

A	First matrix.
B	Second matrix.

Returns

Result of stacking first matrix on top of second.

Definition at line 117 of file append_row.hpp.

template<typename T , int R1, int R2>

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::append_row ( const Eigen::Matrix< T, R1, 1 > &  A, const Eigen::Matrix< T, R2, 1 > &  B  )

inline

Return the result of stacking the first vector on top of the second vector, with the result being a vector.

This is an overloaded template function for the case where both inputs have the same scalar type.

This function applies to (vector, vector) and returns a vector.

Template Parameters

T	type of elements in both vectors
R1	number of rows in the first vector, can be Eigen::Dynamic
R2	number of rows in the second vector, can be Eigen::Dynamic

Parameters

A	First vector.
B	Second vector.

Returns

Result of stacking first vector on top of the second vector.

Definition at line 148 of file append_row.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, 1> stan::math::append_row ( const T1 &  A, const Eigen::Matrix< T2, R, C > &  B  )

inline

Return the result of stacking an scalar on top of the a vector, with the result being a vector.

This function applies to (scalar, vector) and returns a vector.

Template Parameters

T1	type of the scalar
T2	type of elements in the vector
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

A	scalar.
B	vector.

Returns

Result of stacking the scalar on top of the vector.

Definition at line 174 of file append_row.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, 1> stan::math::append_row ( const Eigen::Matrix< T1, R, C > &  A, const T2 &  B  )

inline

Return the result of stacking a vector on top of the an scalar, with the result being a vector.

This function applies to (vector, scalar) and returns a vector.

Template Parameters

T1	type of elements in the vector
T2	type of the scalar
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

A	vector.
B	scalar.

Returns

Result of stacking the vector on top of the scalar.

Definition at line 201 of file append_row.hpp.

template<typename T , typename = require_eigen_t<T>>

auto stan::math::as_array_or_scalar ( const T &  v )

inline

Converts a matrix type to an array.

Template Parameters

T	Type of Eigen Matrix or expression

Parameters

v	Specified Eigen Matrix or expression.

Returns

Matrix converted to an array.

Definition at line 31 of file as_array_or_scalar.hpp.

int stan::math::as_bool ( const var &  v )

inline

Return 1 if the argument is unequal to zero and 0 otherwise.

Parameters

v

Value.

Returns

1 if argument is equal to zero (or NaN) and 0 otherwise.

Definition at line 16 of file as_bool.hpp.

template<typename T >

bool stan::math::as_bool ( const T &  x )

inline

Return true if the argument is not equal to zero (in the != operator sense) and false otherwise.

Template Parameters

T	type of scalar

Parameters

x

value

Returns

true if value is not equal to zero

Definition at line 17 of file as_bool.hpp.

template<typename T_operation , typename = std::enable_if_t<std::is_base_of< operation_cl_base, std::remove_reference_t<T_operation>>::value>>

T_operation&& stan::math::as_operation_cl ( T_operation &&  a )

inline

Converts any valid kernel generator expression into an operation.

This is an overload for operations - a no-op

Template Parameters

T_operation

type of the input operation

Parameters

a	an operation

Returns

operation

Definition at line 26 of file as_operation_cl.hpp.

template<typename T_scalar , typename = require_arithmetic_t<T_scalar>>

scalar_<T_scalar> stan::math::as_operation_cl ( const T_scalar  a )

inline

Converts any valid kernel generator expression into an operation.

This is an overload for scalars (arithmetic types). It wraps them into scalar_.

Template Parameters

T_scalar

type of the input scalar

Parameters

a

scalar

Returns

scalar_ wrapping the input

Definition at line 38 of file as_operation_cl.hpp.

template<typename T_matrix_cl , typename = require_matrix_cl_t<T_matrix_cl>>

load_<T_matrix_cl> stan::math::as_operation_cl ( T_matrix_cl &&  a )

inline

Converts any valid kernel generator expression into an operation.

This is an overload for matrix_cl. It wraps them into into load_.

Template Parameters

T_matrix_cl

matrix_cl

Parameters

a

matrix_cl

Returns

load_ wrapping the input

Definition at line 50 of file as_operation_cl.hpp.

template<typename T >

fvar<T> stan::math::asin ( const fvar< T > &  x )

inline

Definition at line 13 of file asin.hpp.

template<typename T >

apply_scalar_unary<asin_fun, T>::return_t stan::math::asin ( const T &  x )

inline

Vectorized version of asin().

Template Parameters

T	type of container

Parameters

x

container

Returns

Arcsine of each variable in the container, in radians.

Definition at line 33 of file asin.hpp.

var stan::math::asin ( const var &  a )

inline

Return the principal value of the arc sine, in radians, of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}\).

\[ \mbox{asin}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \arcsin(x) & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asin}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \arcsin(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x < -1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arcsin(x)}{\partial x} = \frac{1}{\sqrt{1-x^2}} \]

Parameters

a	Variable in range [-1, 1].

Returns

Arc sine of variable, in radians.

Definition at line 57 of file asin.hpp.

template<typename T >

fvar<T> stan::math::asinh ( const fvar< T > &  x )

inline

Definition at line 14 of file asinh.hpp.

double stan::math::asinh ( double  x )

inline

Return the inverse hyperbolic sine of the specified value.

Returns infinity for infinity argument and -infinity for -infinity argument. Returns nan for nan argument.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic sine of the argument.

Definition at line 19 of file asinh.hpp.

double stan::math::asinh ( int  x )

inline

Integer version of asinh.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic sine of the argument.

Definition at line 27 of file asinh.hpp.

template<typename T >

apply_scalar_unary<asinh_fun, T>::return_t stan::math::asinh ( const T &  x )

inline

Vectorized version of asinh().

Template Parameters

T	type of container

Parameters

x

container

Returns

Inverse hyperbolic sine of each value in the container.

Definition at line 32 of file asinh.hpp.

var stan::math::asinh ( const var &  a )

inline

The inverse hyperbolic sine function for variables (C99).

The derivative is defined by

\(\frac{d}{dx} \mbox{asinh}(x) = \frac{x}{x^2 + 1}\).

\[ \mbox{asinh}(x) = \begin{cases} \sinh^{-1}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asinh}(x)}{\partial x} = \begin{cases} \frac{\partial\, \sinh^{-1}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right) \]

\[ \frac{\partial \, \sinh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2+1}} \]

Parameters

a	The variable.

Returns

Inverse hyperbolic sine of the variable.

Definition at line 55 of file asinh.hpp.

template<typename T_lhs , typename T_rhs >

void stan::math::assign ( T_lhs &  x, const T_rhs &  y  )

inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will match arguments where the right-hand side is assignable to the left and they are not both std::vector or Eigen::Matrix types.

Template Parameters

T_lhs	Type of left-hand side.
T_rhs	Type of right-hand side.

Parameters

x	Left-hand side.
y	Right-hand side.

Definition at line 45 of file assign.hpp.

template<typename T_lhs , typename T_rhs , int R1, int C1, int R2, int C2>

void stan::math::assign ( Eigen::Matrix< T_lhs, R1, C1 > &  x, const Eigen::Matrix< T_rhs, R2, C2 > &  y  )

inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both Eigen::Matrix types, but whose shapes are not compatible. The shapes are specified in the row and column template parameters.

Template Parameters

T_lhs	Type of left-hand side matrix elements.
T_rhs	Type of right-hand side matrix elements.
R1	number of rows in the left-hand side matrix or Eigen::Dynamic
C1	number of columns in the left-hand side matrix or Eigen::Dynamic
R2	number of rows in the right-hand side matrix or Eigen::Dynamic
C2	number of columns in the right-hand side matrix or Eigen::Dynamic

Parameters

x	Left-hand side matrix.
y	Right-hand side matrix.

Exceptions

std::invalid_argument

Definition at line 71 of file assign.hpp.

template<typename T_lhs , typename T_rhs , int R, int C>

void stan::math::assign ( Eigen::Matrix< T_lhs, R, C > &  x, const Eigen::Matrix< T_rhs, R, C > &  y  )

inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both Eigen::Matrix types and whose shapes match. The shapes are specified in the row and column template parameters.

Template Parameters

T_lhs	Type of left-hand side matrix elements.
T_rhs	Type of right-hand side matrix elements.
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

x	Left-hand side matrix.
y	Right-hand side matrix.

Exceptions

std::invalid_argument

if sizes do not match.

Definition at line 107 of file assign.hpp.

template<typename T_lhs , typename T , int R, int C>

void stan::math::assign ( Eigen::Block< T_lhs >  x, const Eigen::Matrix< T, R, C > &  y  )

inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both Eigen::Matrix types and whose shapes match. The shape of the right-hand side matrix is specified in the row and column shape template parameters.

The left-hand side is intentionally not a reference, because that won't match generally enough; instead, a non-reference is used, which still holds onto a reference to the contained matrix and thus still updates the appropriate values.

Template Parameters

T_lhs	Type of matrix block elements.
T	Type of right-hand side matrix elements.
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

x	Left-hand side block view of matrix.
y	Right-hand side matrix.

Exceptions

std::invalid_argument

if sizes do not match.

Definition at line 140 of file assign.hpp.

template<typename T_lhs , typename T_rhs >

void stan::math::assign ( std::vector< T_lhs > &  x, const std::vector< T_rhs > &  y  )

inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both std::vector, and will call assign() element-by element.

For example, a std::vector<int> can be assigned to a std::vector<double> using this function.

Template Parameters

T_lhs	Type of left-hand side vector elements.
T_rhs	Type of right-hand side vector elements.

Parameters

x	Left-hand side vector.
y	Right-hand side vector.

Exceptions

std::invalid_argument

if sizes do not match.

Definition at line 172 of file assign.hpp.

template<typename T >

fvar<T> stan::math::atan ( const fvar< T > &  x )

inline

Definition at line 13 of file atan.hpp.

template<typename T >

apply_scalar_unary<atan_fun, T>::return_t stan::math::atan ( const T &  x )

inline

Vectorized version of atan().

Template Parameters

T	type of container

Parameters

x

container

Returns

Arctan of each value in x, in radians.

Definition at line 33 of file atan.hpp.

var stan::math::atan ( const var &  a )

inline

Return the principal value of the arc tangent, in radians, of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \arctan x = \frac{1}{1 + x^2}\).

\[ \mbox{atan}(x) = \begin{cases} \arctan(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan}(x)}{\partial x} = \begin{cases} \frac{\partial\, \arctan(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arctan(x)}{\partial x} = \frac{1}{x^2+1} \]

Parameters

a	Variable in range [-1, 1].

Returns

Arc tangent of variable, in radians.

Definition at line 49 of file atan.hpp.

template<typename T >

fvar<T> stan::math::atan2 ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 13 of file atan2.hpp.

template<typename T >

fvar<T> stan::math::atan2 ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 21 of file atan2.hpp.

template<typename T >

fvar<T> stan::math::atan2 ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 28 of file atan2.hpp.

var stan::math::atan2 ( const var &  a, const var &  b  )

inline

Return the principal value of the arc tangent, in radians, of the first variable divided by the second (cmath).

The partial derivatives are defined by

\( \frac{\partial}{\partial x} \arctan \frac{x}{y} = \frac{y}{x^2 + y^2}\), and

\( \frac{\partial}{\partial y} \arctan \frac{x}{y} = \frac{-x}{x^2 + y^2}\).

Parameters

a	Numerator variable.
b	Denominator variable.

Returns

The arc tangent of the fraction, in radians.

Definition at line 61 of file atan2.hpp.

var stan::math::atan2 ( const var &  a, double  b  )

inline

Return the principal value of the arc tangent, in radians, of the first variable divided by the second scalar (cmath).

The derivative with respect to the variable is

\( \frac{d}{d x} \arctan \frac{x}{c} = \frac{c}{x^2 + c^2}\).

Parameters

a	Numerator variable.
b	Denominator scalar.

Returns

The arc tangent of the fraction, in radians.

Definition at line 77 of file atan2.hpp.

var stan::math::atan2 ( double  a, const var &  b  )

inline

Return the principal value of the arc tangent, in radians, of the first scalar divided by the second variable (cmath).

The derivative with respect to the variable is

\( \frac{\partial}{\partial y} \arctan \frac{c}{y} = \frac{-c}{c^2 + y^2}\).

\[ \mbox{atan2}(x, y) = \begin{cases} \arctan\left(\frac{x}{y}\right) & \mbox{if } -\infty\leq x \leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan2}(x, y)}{\partial x} = \begin{cases} \frac{y}{x^2+y^2} & \mbox{if } -\infty\leq x\leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan2}(x, y)}{\partial y} = \begin{cases} -\frac{x}{x^2+y^2} & \mbox{if } -\infty\leq x\leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	Numerator scalar.
b	Denominator variable.

Returns

The arc tangent of the fraction, in radians.

Definition at line 117 of file atan2.hpp.

template<typename T >

fvar<T> stan::math::atanh ( const fvar< T > &  x )

inline

Return inverse hyperbolic tangent of specified value.

Template Parameters

T	scalar type of forward-mode autodiff variable argument.

Parameters

x

Argument.

Returns

Inverse hyperbolic tangent of argument.

Exceptions

std::domain_error

if x < -1 or x > 1.

Definition at line 22 of file atanh.hpp.

double stan::math::atanh ( double  x )

inline

Return the inverse hyperbolic tangent of the specified value.

An argument of -1 returns negative infinity and an argument of 1 returns infinity. Returns nan for nan argument.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic tangent of the argument.

Exceptions

std::domain_error

If argument is not in [-1, 1].

Definition at line 22 of file atanh.hpp.

template<typename T >

apply_scalar_unary<atanh_fun, T>::return_t stan::math::atanh ( const T &  x )

inline

Return the elementwise application of atanh() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise atanh of members of container.

Definition at line 38 of file atanh.hpp.

double stan::math::atanh ( int  x )

inline

Integer version of atanh.

Parameters

[in]

x

Argument.

Returns

Inverse hyperbolic tangent of the argument.

Exceptions

std::domain_error

If argument is less than 1.

Definition at line 38 of file atanh.hpp.

var stan::math::atanh ( const var &  a )

inline

The inverse hyperbolic tangent function for variables (C99).

The derivative is defined by

\(\frac{d}{dx} \mbox{atanh}(x) = \frac{1}{1 - x^2}\).

\[ \mbox{atanh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \tanh^{-1}(x) & \mbox{if } -1\leq x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atanh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \tanh^{-1}(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \tanh^{-1}(x)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \]

\[ \frac{\partial \, \tanh^{-1}(x)}{\partial x} = \frac{1}{1-x^2} \]

Parameters

a	The variable.

Returns

Inverse hyperbolic tangent of the variable.

Exceptions

std::domain_error

if a < -1 or a > 1

Definition at line 58 of file atanh.hpp.

template<typename T >

void stan::math::autocorrelation	(	const std::vector< T > &	y,
		std::vector< T > &	ac,
		Eigen::FFT< T > &	fft
	)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
ac	Autocorrelations.
fft	FFT engine instance.

Definition at line 61 of file autocorrelation.hpp.

template<typename T , typename DerivedA , typename DerivedB >

void stan::math::autocorrelation	(	const Eigen::MatrixBase< DerivedA > &	y,
		Eigen::MatrixBase< DerivedB > &	ac,
		Eigen::FFT< T > &	fft
	)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
ac	Autocorrelations.
fft	FFT engine instance.

Definition at line 118 of file autocorrelation.hpp.

template<typename T >

void stan::math::autocorrelation	(	const std::vector< T > &	y,
		std::vector< T > &	ac
	)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocorrelation function.

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
ac	Autocorrelations.

Definition at line 163 of file autocorrelation.hpp.

template<typename T , typename DerivedA , typename DerivedB >

void stan::math::autocorrelation	(	const Eigen::MatrixBase< DerivedA > &	y,
		Eigen::MatrixBase< DerivedB > &	ac
	)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocorrelation function

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
ac	Autocorrelations.

Definition at line 190 of file autocorrelation.hpp.

template<typename T >

void stan::math::autocovariance	(	const std::vector< T > &	y,
		std::vector< T > &	acov,
		Eigen::FFT< T > &	fft
	)

Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
acov	Autocovariance.
fft	FFT engine instance.

Definition at line 33 of file autocovariance.hpp.

template<typename T , typename DerivedA , typename DerivedB >

void stan::math::autocovariance	(	const Eigen::MatrixBase< DerivedA > &	y,
		Eigen::MatrixBase< DerivedB > &	acov,
		Eigen::FFT< T > &	fft
	)

Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
acov	Autocovariance.
fft	FFT engine instance.

Definition at line 64 of file autocovariance.hpp.

template<typename T >

void stan::math::autocovariance	(	const std::vector< T > &	y,
		std::vector< T > &	acov
	)

Write autocovariance estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocovariance function.

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
acov	Autocovariances.

Definition at line 87 of file autocovariance.hpp.

template<typename T , typename DerivedA , typename DerivedB >

void stan::math::autocovariance	(	const Eigen::MatrixBase< DerivedA > &	y,
		Eigen::MatrixBase< DerivedB > &	acov
	)

Write autocovariance estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocovariance function

Template Parameters

T	Scalar type.

Parameters

y	Input sequence.
acov	Autocovariances.

Definition at line 114 of file autocovariance.hpp.

template<typename T_alpha , typename T_beta >

return_type_t<T_beta, T_alpha> stan::math::bernoulli_logit_glm_lpmf ( const matrix_cl< int > &  y, const matrix_cl< double > &  x, const T_alpha &  alpha, const T_beta &  beta  )

inline

Definition at line 145 of file bernoulli_logit_glm_lpmf.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta >

return_type_t<T_x, T_beta, T_alpha> stan::math::bernoulli_logit_glm_lpmf ( const T_y &  y, const T_x &  x, const T_alpha &  alpha, const T_beta &  beta  )

inline

Definition at line 168 of file bernoulli_logit_glm_lpmf.hpp.

template<typename T_n , typename T_prob >

return_type_t<T_prob> stan::math::bernoulli_logit_lpmf ( const T_n &  n, const T_prob &  theta  )

inline

Definition at line 85 of file bernoulli_logit_lpmf.hpp.

template<typename T_y , typename T_prob >

return_type_t<T_prob> stan::math::bernoulli_lpmf ( const T_y &  n, const T_prob &  theta  )

inline

Definition at line 107 of file bernoulli_lpmf.hpp.

template<typename T >

fvar<T> stan::math::bessel_first_kind ( int  v, const fvar< T > &  z  )

inline

Definition at line 12 of file bessel_first_kind.hpp.

var stan::math::bessel_first_kind ( int  v, const var &  a  )

inline

Definition at line 25 of file bessel_first_kind.hpp.

template<typename T2 >

T2 stan::math::bessel_first_kind ( int  v, const T2  z  )

inline

\[ \mbox{bessel\_first\_kind}(v, x) = \begin{cases} J_v(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{bessel\_first\_kind}(v, x)}{\partial x} = \begin{cases} \frac{\partial\, J_v(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ J_v(x)=\left(\frac{1}{2}x\right)^v \sum_{k=0}^\infty \frac{\left(-\frac{1}{4}x^2\right)^k}{k!\, \Gamma(v+k+1)} \]

\[ \frac{\partial \, J_v(x)}{\partial x} = \frac{v}{x}J_v(x)-J_{v+1}(x) \]

Definition at line 38 of file bessel_first_kind.hpp.

template<typename T >

fvar<T> stan::math::bessel_second_kind ( int  v, const fvar< T > &  z  )

inline

Definition at line 12 of file bessel_second_kind.hpp.

var stan::math::bessel_second_kind ( int  v, const var &  a  )

inline

Definition at line 25 of file bessel_second_kind.hpp.

template<typename T2 >

T2 stan::math::bessel_second_kind ( int  v, const T2  z  )

inline

\[ \mbox{bessel\_second\_kind}(v, x) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ Y_v(x) & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{bessel\_second\_kind}(v, x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ \frac{\partial\, Y_v(x)}{\partial x} & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ Y_v(x)=\frac{J_v(x)\cos(v\pi)-J_{-v}(x)}{\sin(v\pi)} \]

\[ \frac{\partial \, Y_v(x)}{\partial x} = \frac{v}{x}Y_v(x)-Y_{v+1}(x) \]

Definition at line 40 of file bessel_second_kind.hpp.

template<typename T >

fvar<T> stan::math::beta ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return fvar with the beta function applied to the specified arguments and its gradient.

The beta function is defined for \(a > 0\) and \(b > 0\) by

\(\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\).

\[ \mbox{beta}(\alpha, \beta) = \begin{cases} \int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \left(\psi(\alpha)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \left(\psi(\beta)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Template Parameters

T	Type of values.

Parameters

x1	First value
x2	Second value

Returns

Fvar with result beta function of arguments and gradients.

Definition at line 51 of file beta.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::beta ( const T1  a, const T2  b  )

inline

Return the beta function applied to the specified arguments.

The beta function is defined for \(a > 0\) and \(b > 0\) by

\(\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\).

\[ \mbox{beta}(\alpha, \beta) = \begin{cases} \int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \left(\psi(\alpha)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \left(\psi(\beta)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Template Parameters

T1	Type of first value.
T2	Type of second value.

Parameters

a	First value
b	Second value

Returns

Beta function applied to the two values.

Definition at line 51 of file beta.hpp.

template<typename T >

fvar<T> stan::math::beta ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 60 of file beta.hpp.

template<typename T >

fvar<T> stan::math::beta ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 67 of file beta.hpp.

var stan::math::beta ( const var &  a, const var &  b  )

inline

Definition at line 69 of file beta.hpp.

var stan::math::beta ( const var &  a, double  b  )

inline

Definition at line 91 of file beta.hpp.

var stan::math::beta ( double  a, const var &  b  )

inline

Definition at line 113 of file beta.hpp.

template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >

return_type_t<T_size1, T_size2> stan::math::beta_binomial_lpmf	(	const T_n &	n,
		const T_N &	N,
		const T_size1 &	alpha,
		const T_size2 &	beta
	)

Definition at line 146 of file beta_binomial_lpmf.hpp.

template<typename T_y , typename T_scale_succ , typename T_scale_fail >

return_type_t<T_y, T_scale_succ, T_scale_fail> stan::math::beta_lpdf ( const T_y &  y, const T_scale_succ &  alpha, const T_scale_fail &  beta  )

inline

Definition at line 180 of file beta_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_prec >

return_type_t<T_y, T_loc, T_prec> stan::math::beta_proportion_lpdf ( const T_y &  y, const T_loc &  mu, const T_prec &  kappa  )

inline

Definition at line 171 of file beta_proportion_lpdf.hpp.

template<typename T >

fvar<T> stan::math::binary_log_loss ( int  y, const fvar< T > &  y_hat  )

inline

Definition at line 12 of file binary_log_loss.hpp.

template<typename T >

T stan::math::binary_log_loss ( int  y, const T &  y_hat  )

inline

Returns the log loss function for binary classification with specified reference and response values.

The log loss function for prediction \(\hat{y} \in [0, 1]\) given outcome \(y \in \{ 0, 1 \}\) is

\(\mbox{logloss}(1, \hat{y}) = -\log \hat{y} \), and

\(\mbox{logloss}(0, \hat{y}) = -\log (1 - \hat{y}) \).

Template Parameters

T	value type

Parameters

[in]	y	reference value, either 0 or 1
[in]	y_hat	response value in [0, 1]

Returns

Log loss for response given reference value

Definition at line 28 of file binary_log_loss.hpp.

var stan::math::binary_log_loss ( int  y, const var &  y_hat  )

inline

The log loss function for variables (stan).

See binary_log_loss() for the double-based version.

The derivative with respect to the variable \(\hat{y}\) is

\(\frac{d}{d\hat{y}} \mbox{logloss}(1, \hat{y}) = - \frac{1}{\hat{y}}\), and

\(\frac{d}{d\hat{y}} \mbox{logloss}(0, \hat{y}) = \frac{1}{1 - \hat{y}}\).

\[ \mbox{binary\_log\_loss}(y, \hat{y}) = \begin{cases} y \log \hat{y} + (1 - y) \log (1 - \hat{y}) & \mbox{if } 0\leq \hat{y}\leq 1, y\in\{ 0, 1 \}\\[6pt] \textrm{NaN} & \mbox{if } \hat{y} = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binary\_log\_loss}(y, \hat{y})}{\partial \hat{y}} = \begin{cases} \frac{y}{\hat{y}}-\frac{1-y}{1-\hat{y}} & \mbox{if } 0\leq \hat{y}\leq 1, y\in\{ 0, 1 \}\\[6pt] \textrm{NaN} & \mbox{if } \hat{y} = \textrm{NaN} \end{cases} \]

Parameters

y	Reference value.
y_hat	Response variable.

Returns

Log loss of response versus reference value.

Definition at line 62 of file binary_log_loss.hpp.

template<typename T_N , typename T_n >

return_type_t<T_N, T_n> stan::math::binomial_coefficient_log ( const T_N  N, const T_n  n  )

inline

Return the log of the binomial coefficient for the specified arguments.

The binomial coefficient, \({N \choose n}\), read "N choose n", is defined for \(0 \leq n \leq N\) by

\({N \choose n} = \frac{N!}{n! (N-n)!}\).

This function uses Gamma functions to define the log and generalize the arguments to continuous N and n.

\( \log {N \choose n} = \log \ \Gamma(N+1) - \log \Gamma(n+1) - \log \Gamma(N-n+1)\).

\[ \mbox{binomial\_coefficient\_log}(x, y) = \begin{cases} \textrm{error} & \mbox{if } y > x \textrm{ or } y < 0\\ \ln\Gamma(x+1) & \mbox{if } 0\leq y \leq x \\ \quad -\ln\Gamma(y+1)& \\ \quad -\ln\Gamma(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binomial\_coefficient\_log}(x, y)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } y > x \textrm{ or } y < 0\\ \Psi(x+1) & \mbox{if } 0\leq y \leq x \\ \quad -\Psi(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binomial\_coefficient\_log}(x, y)}{\partial y} = \begin{cases} \textrm{error} & \mbox{if } y > x \textrm{ or } y < 0\\ -\Psi(y+1) & \mbox{if } 0\leq y \leq x \\ \quad +\Psi(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

N	total number of objects.
n	number of objects chosen.

Returns

log (N choose n).

Definition at line 62 of file binomial_coefficient_log.hpp.

template<typename T_n , typename T_N , typename T_prob >

return_type_t<T_prob> stan::math::binomial_logit_lpmf ( const T_n &  n, const T_N &  N, const T_prob &  alpha  )

inline

Definition at line 111 of file binomial_logit_lpmf.hpp.

template<typename T_n , typename T_N , typename T_prob >

return_type_t<T_prob> stan::math::binomial_lpmf ( const T_n &  n, const T_N &  N, const T_prob &  theta  )

inline

Definition at line 105 of file binomial_lpmf.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::block ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m, size_t  i, size_t  j, size_t  nrows, size_t  ncols  )

inline

Return a nrows x ncols submatrix starting at (i-1, j-1).

Template Parameters

T	type of elements in the matrix

Parameters

m	Matrix.
i	Starting row.
j	Starting column.
nrows	Number of rows in block.
ncols	Number of columns in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 22 of file block.hpp.

template<int R, int C>

vari** stan::math::build_vari_array

(

const Eigen::Matrix< var, R, C > &

x

)

Allocates and populates a flat array of vari pointers in the autodiff arena with the varis pointed to by the vars in the input Eigen matrix.

Template Parameters

R	Eigen row type of x
C	Eigen column type of x

Parameters

x

Input

Returns

Flat array of vari pointers

Definition at line 21 of file build_vari_array.hpp.

double stan::math::calculate_chain ( double  x, double  val  )

inline

Definition at line 10 of file calculate_chain.hpp.

template<typename T_alpha_scalar , typename T_beta_scalar >

return_type_t<T_alpha_scalar, T_beta_scalar> stan::math::categorical_logit_glm_lpmf	(	const matrix_cl< int > &	y,
		const matrix_cl< double > &	x,
		const Eigen::Matrix< T_alpha_scalar, Eigen::Dynamic, 1 > &	alpha,
		const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, Eigen::Dynamic > &	beta
	)

Definition at line 142 of file categorical_logit_glm_lpmf.hpp.

template<typename T_y , typename T_x_scalar , int T_x_rows, typename T_alpha_scalar , typename T_beta_scalar >

return_type_t<T_x_scalar, T_alpha_scalar, T_beta_scalar> stan::math::categorical_logit_glm_lpmf	(	const T_y &	y,
		const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &	x,
		const Eigen::Matrix< T_alpha_scalar, Eigen::Dynamic, 1 > &	alpha,
		const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, Eigen::Dynamic > &	beta
	)

Definition at line 187 of file categorical_logit_glm_lpmf.hpp.

template<bool propto, typename T_prob >

return_type_t<T_prob> stan::math::categorical_logit_lpmf	(	int	n,
		const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &	beta
	)

Definition at line 17 of file categorical_logit_lpmf.hpp.

template<typename T_prob >

return_type_t<T_prob> stan::math::categorical_logit_lpmf ( int  n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  beta  )

inline

Definition at line 34 of file categorical_logit_lpmf.hpp.

template<bool propto, typename T_prob >

return_type_t<T_prob> stan::math::categorical_logit_lpmf	(	const std::vector< int > &	ns,
		const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &	beta
	)

Definition at line 40 of file categorical_logit_lpmf.hpp.

template<typename T_prob >

return_type_t<T_prob> stan::math::categorical_logit_lpmf ( const std::vector< int > &  ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  beta  )

inline

Definition at line 70 of file categorical_logit_lpmf.hpp.

template<bool propto, typename T_prob >

return_type_t<T_prob> stan::math::categorical_lpmf	(	int	n,
		const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &	theta
	)

Definition at line 15 of file categorical_lpmf.hpp.

template<typename T_prob >

return_type_t<T_prob> stan::math::categorical_lpmf ( const typename math::index_type< Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > >::type  n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta  )

inline

Definition at line 33 of file categorical_lpmf.hpp.

template<bool propto, typename T_prob >

return_type_t<T_prob> stan::math::categorical_lpmf	(	const std::vector< int > &	ns,
		const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &	theta
	)

Definition at line 41 of file categorical_lpmf.hpp.

template<typename T_prob >

return_type_t<T_prob> stan::math::categorical_lpmf ( const std::vector< int > &  ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta  )

inline

Definition at line 80 of file categorical_lpmf.hpp.

template<class RNG >

int stan::math::categorical_rng ( const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  theta, RNG &  rng  )

inline

Definition at line 14 of file categorical_rng.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::cauchy_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 113 of file cauchy_lpdf.hpp.

double stan::math::cbrt ( double  x )

inline

Return the cube root of the specified value.

Parameters

[in]

x

Argument.

Returns

Cube root of the argument.

Exceptions

std::domain_error

If argument is negative.

Definition at line 17 of file cbrt.hpp.

template<typename T >

fvar<T> stan::math::cbrt ( const fvar< T > &  x )

inline

Return cube root of specified argument.

Template Parameters

T	Scalar type of autodiff variable.

Parameters

x

Argument.

Returns

Cube root of argument.

Definition at line 20 of file cbrt.hpp.

double stan::math::cbrt ( int  x )

inline

Integer version of cbrt.

Parameters

[in]

x

Argument.

Returns

Cube root of the argument.

Exceptions

std::domain_error

If argument is less than 1.

Definition at line 26 of file cbrt.hpp.

template<typename T >

apply_scalar_unary<cbrt_fun, T>::return_t stan::math::cbrt ( const T &  x )

inline

Vectorized version of cbrt().

Template Parameters

T	type of container

Parameters

x

container

Returns

Cube root of each value in x.

Definition at line 32 of file cbrt.hpp.

var stan::math::cbrt ( const var &  a )

inline

Returns the cube root of the specified variable (C99).

The derivative is

\(\frac{d}{dx} x^{1/3} = \frac{1}{3 x^{2/3}}\).

\[ \mbox{cbrt}(x) = \begin{cases} \sqrt[3]{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cbrt}(x)}{\partial x} = \begin{cases} \frac{1}{3x^{2/3}} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Specified variable.

Returns

Cube root of the variable.

Definition at line 45 of file cbrt.hpp.

template<typename T >

fvar<T> stan::math::ceil ( const fvar< T > &  x )

inline

Definition at line 12 of file ceil.hpp.

template<typename T >

apply_scalar_unary<ceil_fun, T>::return_t stan::math::ceil ( const T &  x )

inline

Vectorized version of ceil().

Template Parameters

T	type of container

Parameters

x

container

Returns

Least integer >= each value in x.

Definition at line 33 of file ceil.hpp.

var stan::math::ceil ( const var &  a )

inline

Return the ceiling of the specified variable (cmath).

The derivative of the ceiling function is defined and zero everywhere but at integers, and we set them to zero for convenience,

\(\frac{d}{dx} {\lceil x \rceil} = 0\).

The ceiling function rounds up. For double values, this is the smallest integral value that is not less than the specified value. Although this function is not differentiable because it is discontinuous at integral values, its gradient is returned as zero everywhere.

\[ \mbox{ceil}(x) = \begin{cases} \lceil x\rceil & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{ceil}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Input variable.

Returns

Ceiling of the variable.

Definition at line 59 of file ceil.hpp.

template<typename T_a1 , typename T_a2 , typename T_b1 , typename T_z >

void stan::math::check_2F1_converges ( const char *  function, const T_a1 &  a1, const T_a2 &  a2, const T_b1 &  b1, const T_z &  z  )

inline

Check if the hypergeometric function (2F1) called with supplied arguments will converge, assuming arguments are finite values.

Template Parameters

T_a1	Type of a1
T_a2	Type of a2
T_b1	Type of b1
T_z	Type of z

Parameters

function	Name of function ultimately relying on 2F1 (for error messages)
a1	Variable to check
a2	Variable to check
b1	Variable to check
z	Variable to check

Exceptions

<code>domain_error</code>

if 2F1(a1, a2, b1, z) does not meet convergence conditions, or if any coefficient is NaN.

Definition at line 33 of file check_2F1_converges.hpp.

template<typename T_a1 , typename T_a2 , typename T_a3 , typename T_b1 , typename T_b2 , typename T_z >

void stan::math::check_3F2_converges ( const char *  function, const T_a1 &  a1, const T_a2 &  a2, const T_a3 &  a3, const T_b1 &  b1, const T_b2 &  b2, const T_z &  z  )

inline

Check if the hypergeometric function (3F2) called with supplied arguments will converge, assuming arguments are finite values.

Template Parameters

T_a1	Type of a1
T_a2	Type of a2
T_a3	Type of a3
T_b1	Type of b1
T_b2	Type of b2
T_z	Type of z

Parameters

function	Name of function ultimately relying on 3F2 (for error & messages)
a1	Variable to check
a2	Variable to check
a3	Variable to check
b1	Variable to check
b2	Variable to check
z	Variable to check

Exceptions

<code>domain_error</code>

if 3F2(a1, a2, a3, b1, b2, z) does not meet convergence conditions, or if any coefficient is NaN.

Definition at line 38 of file check_3F2_converges.hpp.

template<typename T_y , typename T_low , typename T_high >

void stan::math::check_bounded ( const char *  function, const char *  name, const T_y &  y, const T_low &  low, const T_high &  high  )

inline

Check if the value is between the low and high values, inclusively.

Template Parameters

T_y	Type of value
T_low	Type of low value
T_high	Type of high value

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Value to check
low	Low bound
high	High bound

Exceptions

<code>std::domain_error</code>

otherwise. This also throws if any of the arguments are NaN.

Definition at line 70 of file check_bounded.hpp.

template<typename T_y >

void stan::math::check_cholesky_factor ( const char *  function, const char *  name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y  )

inline

Check if the specified matrix is a valid Cholesky factor.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. Note that Cholesky factors need not be square, but require at least as many rows M as columns N (i.e., M >= N).

Template Parameters

T_y	Type of elements of Cholesky factor

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::domain_error</code>

if y is not a valid Choleksy factor, if number of rows is less than the number of columns, if there are 0 columns, or if any element in matrix is NaN

Definition at line 28 of file check_cholesky_factor.hpp.

template<typename T_y >

void stan::math::check_cholesky_factor_corr	(	const char *	function,
		const char *	name,
		const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &	y
	)

Check if the specified matrix is a valid Cholesky factor of a correlation matrix.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. Note that Cholesky factors need not be square, but requires at least as many rows M as columns N (i.e., M >= N). Tolerance is specified by math::CONSTRAINT_TOLERANCE.

Template Parameters

T_y	Type of elements of Cholesky factor

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::domain_error</code>

if y is not a valid Choleksy factor, if number of rows is less than the number of columns, if there are 0 columns, or if any element in matrix is NaN

Definition at line 31 of file check_cholesky_factor_corr.hpp.

template<typename T_y , int R, int C>

void stan::math::check_column_index ( const char *  function, const char *  name, const Eigen::Matrix< T_y, R, C > &  y, size_t  i  )

inline

Check if the specified index is a valid column of the matrix.

By default this is a 1-indexed check (as opposed to 0-indexed). Behavior can be changed by setting stan::error_index::value. This function will throw an std::out_of_range exception if the index is out of bounds.

Template Parameters

T_y	Type of scalar
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	matrix to test
i	column index to check

Exceptions

<code>std::out_of_range</code>

if index is an invalid column

Definition at line 30 of file check_column_index.hpp.

template<typename T >

void stan::math::check_consistent_size ( const char *  function, const char *  name, const T &  x, size_t  expected_size  )

inline

Check if the dimension of x is consistent, which is defined to be expected_size if x is a vector or 1 if x is not a vector.

Template Parameters

T	Type of value

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
x	Variable to check for consistent size
expected_size	Expected size if x is a vector

Exceptions

<code>invalid_argument</code>

if the size is inconsistent

Definition at line 23 of file check_consistent_size.hpp.

template<typename T >

void stan::math::check_consistent_size_mvt ( const char *  function, const char *  name, const T &  x, size_t  expected_size  )

inline

Check if the dimension of x is consistent, which is defined to be expected_size if x is a vector of vectors or 1 if x is a single vector.

Template Parameters

T	Type of value

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
x	Variable to check for consistent size
expected_size	Expected size if x is a vector

Exceptions

<code>invalid_argument</code>

if the size is inconsistent

Definition at line 25 of file check_consistent_size_mvt.hpp.

template<typename T1 , typename T2 >

void stan::math::check_consistent_sizes ( const char *  function, const char *  name1, const T1 &  x1, const char *  name2, const T2 &  x2  )

inline

Check if the dimension of x1 is consistent with x2.

Consistent size is defined as having the same size if vector-like or being a scalar.

Template Parameters

T1	Type of x1
T2	Type of x2

Parameters

function	Function name (for error messages)
name1	Variable name (for error messages)
x1	Variable to check for consistent size
name2	Variable name (for error messages)
x2	Variable to check for consistent size

Exceptions

<code>invalid_argument</code>

if sizes are inconsistent

Definition at line 25 of file check_consistent_sizes.hpp.

template<typename T1 , typename T2 , typename T3 >

void stan::math::check_consistent_sizes ( const char *  function, const char *  name1, const T1 &  x1, const char *  name2, const T2 &  x2, const char *  name3, const T3 &  x3  )

inline

Check if the dimension of x1, x2, and x3 areconsistent.

Consistent size is defined as having the same size if vector-like or being a scalar.

Template Parameters

T1	Type of x1
T2	Type of x2
T3	Type of x3

Parameters

function	Function name (for error messages)
name1	Variable name (for error messages)
x1	Variable to check for consistent size
name2	Variable name (for error messages)
x2	Variable to check for consistent size
name3	Variable name (for error messages)
x3	Variable to check for consistent size

Exceptions

<code>invalid_argument</code>

if sizes are inconsistent

Definition at line 51 of file check_consistent_sizes.hpp.

template<typename T1 , typename T2 , typename T3 , typename T4 >

void stan::math::check_consistent_sizes ( const char *  function, const char *  name1, const T1 &  x1, const char *  name2, const T2 &  x2, const char *  name3, const T3 &  x3, const char *  name4, const T4 &  x4  )

inline

Check if the dimension of x1, x2, x3, and x4are consistent.

Consistent size is defined as having the same size if vector-like or being a scalar.

Template Parameters

T1	Type of x1
T2	Type of x2
T3	Type of x3
T4	Type of x4

Parameters

function	Function name (for error messages)
name1	Variable name (for error messages)
x1	Variable to check for consistent size
name2	Variable name (for error messages)
x2	Variable to check for consistent size
name3	Variable name (for error messages)
x3	Variable to check for consistent size
name4	Variable name (for error messages)
x4	Variable to check for consistent size

Exceptions

<code>invalid_argument</code>

if sizes are inconsistent

Definition at line 83 of file check_consistent_sizes.hpp.

template<typename T1 , typename T2 , typename T3 , typename T4 , typename T5 >

void stan::math::check_consistent_sizes ( const char *  function, const char *  name1, const T1 &  x1, const char *  name2, const T2 &  x2, const char *  name3, const T3 &  x3, const char *  name4, const T4 &  x4, const char *  name5, const T5 &  x5  )

inline

Definition at line 99 of file check_consistent_sizes.hpp.

template<typename T1 , typename T2 >

void stan::math::check_consistent_sizes_mvt ( const char *  function, const char *  name1, const T1 &  x1, const char *  name2, const T2 &  x2  )

inline

Check if the dimension of x1 is consistent with x2.

Consistent size is defined as having the same size if vector of vectors or being a single vector.

Template Parameters

T1	Type of x1
T2	Type of x2

Parameters

function	Function name (for error messages)
name1	Variable name (for error messages)
x1	Variable to check for consistent size
name2	Variable name (for error messages)
x2	Variable to check for consistent size

Exceptions

<code>invalid_argument</code>

if sizes are inconsistent

Definition at line 25 of file check_consistent_sizes_mvt.hpp.

template<typename T_y >

void stan::math::check_corr_matrix ( const char *  function, const char *  name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y  )

inline

Check if the specified matrix is a valid correlation matrix.

A valid correlation matrix is symmetric, has a unit diagonal (all 1 values), and has all values between -1 and 1 (inclusive). This function throws exceptions if the variable is not a valid correlation matrix.

Template Parameters

T_y	Type of scalar

Parameters

function	Name of the function this was called from
name	Name of the variable
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>	if the matrix is not square
<code>std::domain_error</code>	if the matrix is non-symmetric, diagonals not near 1, not positive definite, or any of the elements nan

Definition at line 33 of file check_corr_matrix.hpp.

template<typename T_y >

void stan::math::check_cov_matrix ( const char *  function, const char *  name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y  )

inline

Check if the specified matrix is a valid covariance matrix.

A valid covariance matrix is a square, symmetric matrix that is positive definite.

Template Parameters

T	Type of scalar.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>	if the matrix is not square or if the matrix is 0x0
<code>std::domain_error</code>	if the matrix is not symmetric, if the matrix is not positive definite, or if any element of the matrix is nan

Definition at line 25 of file check_cov_matrix.hpp.

template<typename T_y >

void stan::math::check_finite ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is finite.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if y is infinity, -infinity, or NaN

Definition at line 47 of file check_finite.hpp.

void stan::math::check_flag_kinsol ( int  flag, long int  max_num_steps  )

inline

Throws an exception message when the function KINSol() (call to the solver) fails.

When the exception is caused by a tuning parameter the user controls, gives a specific error.

Definition at line 27 of file check_flag_sundials.hpp.

void stan::math::check_flag_sundials ( int  flag, const char *  func_name  )

inline

Throws an exception when a Sundial function fails (i.e.

returns a negative flag)

Definition at line 13 of file check_flag_sundials.hpp.

template<typename T_y , typename T_low >

void stan::math::check_greater ( const char *  function, const char *  name, const T_y &  y, const T_low &  low  )

inline

Check if y is strictly greater than low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	Type of y
T_low	Type of lower bound

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound

Exceptions

<code>domain_error</code>

if y is not greater than low or if any element of y or low is NaN.

Definition at line 63 of file check_greater.hpp.

template<typename T_y , typename T_low >

void stan::math::check_greater_or_equal ( const char *  function, const char *  name, const T_y &  y, const T_low &  low  )

inline

Check if y is greater or equal than low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters

T_y	Type of y
T_low	Type of lower bound

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
low	Lower bound

Exceptions

<code>domain_error</code>

if y is not greater or equal to low or if any element of y or low is NaN.

Definition at line 62 of file check_greater_or_equal.hpp.

template<typename T , int R, int C>

void stan::math::check_ldlt_factor ( const char *  function, const char *  name, LDLT_factor< T, R, C > &  A  )

inline

Raise domain error if the specified LDLT factor is invalid.

An LDLT_factor is invalid if it was constructed from a matrix that is not positive definite. The check is that the success() method returns true.

Template Parameters

T	type of scalar
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	function	name of function for error messages
[in]	name	variable name for error messages
[in]	A	the LDLT factor to check for validity

Exceptions

<code>std::domain_error</code>

if the LDLT factor is invalid

Definition at line 27 of file check_ldlt_factor.hpp.

template<typename T_y , typename T_high >

void stan::math::check_less ( const char *  function, const char *  name, const T_y &  y, const T_high &  high  )

inline

Check if y is strictly less than high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	Type of y
T_high	Type of upper bound

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound

Exceptions

<code>domain_error</code>

if y is not less than low or if any element of y or high is NaN.

Definition at line 63 of file check_less.hpp.

template<typename T_y , typename T_high >

void stan::math::check_less_or_equal ( const char *  function, const char *  name, const T_y &  y, const T_high &  high  )

inline

Check if y is less or equal to high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	Type of y
T_high	Type of upper bound

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check
high	Upper bound

Exceptions

<code>std::domain_error</code>

if y is not less than or equal to low or if any element of y or high is NaN

Definition at line 62 of file check_less_or_equal.hpp.

template<typename T_y >

void stan::math::check_lower_triangular ( const char *  function, const char *  name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y  )

inline

Check if the specified matrix is lower triangular.

A matrix x is not lower triangular if there is a non-zero entry x[m, n] with m < n. This function only inspects the upper triangular portion of the matrix, not including the diagonal.

Template Parameters

T	Type of scalar of the matrix

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::domain_error</code>

if the matrix is not lower triangular or if any element in the upper triangular portion is NaN

Definition at line 27 of file check_lower_triangular.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

void stan::math::check_matching_dims ( const char *  function, const char *  name1, const Eigen::Matrix< T1, R1, C1 > &  y1, const char *  name2, const Eigen::Matrix< T2, R2, C2 > &  y2  )

inline

Check if the two matrices are of the same size.

This function checks the runtime sizes only.

Template Parameters

T1	scalar type of the first matrix
T2	scalar type of the second matrix
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

function	name of function (for error messages)
name1	variable name for the first matrix (for error messages)
y1	first matrix to test
name2	variable name for the second matrix (for error messages)
y2	second matrix to test

Exceptions

<code>std::invalid_argument</code>

if the dimensions of the matrices do not match

Definition at line 32 of file check_matching_dims.hpp.

template<bool check_compile, typename T1 , typename T2 , int R1, int C1, int R2, int C2>

void stan::math::check_matching_dims ( const char *  function, const char *  name1, const Eigen::Matrix< T1, R1, C1 > &  y1, const char *  name2, const Eigen::Matrix< T2, R2, C2 > &  y2  )

inline

Check if the two matrices are of the same size.

This function checks the runtime sizes and can also check the static sizes as well. For example, a 4x1 matrix is not the same as a vector with 4 elements.

Template Parameters

check_compile	Whether to check the static sizes
T1	scalar type of the first matrix
T2	scalar type of the second matrix
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

function	name of function (for error messages)
name1	variable name for the first matrix (for error messages)
y1	first matrix to test
name2	variable name for the second matrix (for error messages)
y2	second matrix to test

Exceptions

<code>std::invalid_argument</code>

if the dimensions of the matrices do not match

Definition at line 64 of file check_matching_dims.hpp.

template<typename T_y1 , typename T_y2 >

void stan::math::check_matching_sizes ( const char *  function, const char *  name1, const T_y1 &  y1, const char *  name2, const T_y2 &  y2  )

inline

Check if two structures at the same size.

This function only checks the runtime sizes for variables that implement a size() method.

Template Parameters

T_y1	Type of the first variable
T_y2	Type of the second variable

Parameters

function	Function name (for error messages)
name1	First variable name (for error messages)
y1	First variable
name2	Second variable name (for error messages)
y2	Second variable

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 24 of file check_matching_sizes.hpp.

template<typename T1 , typename T2 >

void stan::math::check_multiplicable ( const char *  function, const char *  name1, const T1 &  y1, const char *  name2, const T2 &  y2  )

inline

Check if the matrices can be multiplied.

This checks the runtime sizes to determine whether the two matrices are multiplicable. This allows Eigen matrices, vectors, and row vectors to be checked.

Template Parameters

T1	Type of first matrix
T2	Type of second matrix

Parameters

function	Function name (for error messages)
name1	Variable name for the first matrix (for error messages)
y1	First matrix to test, requires class access to .rows() and .cols()
name2	Variable name for the second matrix (for error messages)
y2	Second matrix to test, requires class access to .rows() and .cols()

Exceptions

<code>std::invalid_argument</code>

if the matrices are not multiplicable or if either matrix is size 0 for either rows or columns

Definition at line 30 of file check_multiplicable.hpp.

template<typename T_y >

void stan::math::check_nonempty ( const char *  function, const char *  name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y  )

inline

Check if the specified matrix is not empty.

This check does not allow 0x0 matrices.

Template Parameters

T	Type of scalar

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>

if the matrix is empty

Definition at line 22 of file check_nonempty.hpp.

template<typename T_y >

void stan::math::check_nonnegative ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is non-negative.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if y is negative or if any element of y is NaN.

Definition at line 49 of file check_nonnegative.hpp.

template<typename T_y >

void stan::math::check_nonzero_size ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if the specified matrix/vector is of non-zero size.

Throws a std:invalid_argument otherwise. The message will indicate that the variable name "has size 0".

Template Parameters

T_y	Type of container

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Container to test. This will accept matrices and vectors

Exceptions

<code>std::invalid_argument</code>

if the specified matrix/vector has zero size

Definition at line 22 of file check_nonzero_size.hpp.

template<typename T_y >

void stan::math::check_not_nan ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is not NaN.

This function is vectorized and will check each element of y. If any element is NaN, this function will throw an exception.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if any element of y is NaN

Definition at line 48 of file check_not_nan.hpp.

template<typename T_y >

void stan::math::check_ordered	(	const char *	function,
		const char *	name,
		const std::vector< T_y > &	y
	)

Check if the specified vector is sorted into strictly increasing order.

Template Parameters

T_y	Type of scalar

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	std::vector to test

Exceptions

<code>std::domain_error</code>

if the vector elements are not ordered, if there are duplicated values, or if any element is NaN.

Definition at line 24 of file check_ordered.hpp.

template<typename T_y >

void stan::math::check_ordered	(	const char *	function,
		const char *	name,
		const Eigen::Matrix< T_y, Eigen::Dynamic, 1 > &	y
	)

Check if the specified vector is sorted into strictly increasing order.

Template Parameters

T_y	Type of scalar

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Vector to test

Exceptions

<code>std::domain_error</code>

if the vector elements are not ordered, if there are duplicated values, or if any element is NaN.

Definition at line 25 of file check_ordered.hpp.

template<typename T_y >

void stan::math::check_pos_definite ( const char *  function, const char *  name, const Eigen::Matrix< T_y,-1,-1 > &  y  )

inline

Check if the specified square, symmetric matrix is positive definite.

Template Parameters

T_y	Type of scalar of the matrix

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>	if the matrix is not square or if the matrix has 0 size.
<code>std::domain_error</code>	if the matrix is not symmetric, if it is not positive definite, or if any element is NaN

Definition at line 28 of file check_pos_definite.hpp.

template<typename Derived >

void stan::math::check_pos_definite ( const char *  function, const char *  name, const Eigen::LDLT< Derived > &  cholesky  )

inline

Check if the specified LDLT transform of a matrix is positive definite.

Template Parameters

Derived

Derived type of the Eigen::LDLT transform.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
cholesky	Eigen::LDLT to test, whose progenitor must not have any NaN elements

Exceptions

<code>std::domain_error</code>

if the matrix is not positive definite

Definition at line 56 of file check_pos_definite.hpp.

template<typename Derived >

void stan::math::check_pos_definite ( const char *  function, const char *  name, const Eigen::LLT< Derived > &  cholesky  )

inline

Check if the specified LLT decomposition transform resulted in Eigen::Success

Template Parameters

Derived

Derived type of the Eigen::LLT transform.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
cholesky	Eigen::LLT to test, whose progenitor must not have any NaN elements

Exceptions

<code>std::domain_error</code>

if the diagonal of the L matrix is not positive

Definition at line 76 of file check_pos_definite.hpp.

template<typename T_y >

void stan::math::check_pos_semidefinite ( const char *  function, const char *  name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y  )

inline

Check if the specified matrix is positive definite.

Template Parameters

T_y	scalar type of the matrix

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>	if the matrix is not square or if the matrix has 0 size.
<code>std::domain_error</code>	if the matrix is not symmetric, or if it is not positive semi-definite, or if any element of the matrix is NaN.

Definition at line 29 of file check_pos_semidefinite.hpp.

template<typename Derived >

void stan::math::check_pos_semidefinite ( const char *  function, const char *  name, const Eigen::LDLT< Derived > &  cholesky  )

inline

Check if the specified matrix is positive semidefinite.

Template Parameters

Derived

Derived type of the Eigen::LDLT transform.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
cholesky	Eigen::LDLT to test

Exceptions

<code>std::domain_error</code>

if the matrix is not positive semi-definite.

Definition at line 61 of file check_pos_semidefinite.hpp.

template<typename T_y >

void stan::math::check_positive ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is positive.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if y is negative or zero or if any element of y is NaN

Definition at line 54 of file check_positive.hpp.

void stan::math::check_positive ( const char *  function, const char *  name, const char *  expr, int  size  )

inline

Check if size is positive.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
expr	Expression for the dimension size (for error messages)
size	Size value to check

Exceptions

<code>std::invalid_argument</code>

if size is zero or negative.

Definition at line 68 of file check_positive.hpp.

template<typename T_y >

void stan::math::check_positive_finite ( const char *  function, const char *  name, const T_y &  y  )

inline

Check if y is positive and finite.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Variable to check

Exceptions

<code>domain_error</code>

if any element of y is not positive or if any element of y is NaN.

Definition at line 23 of file check_positive_finite.hpp.

template<typename T_y >

void stan::math::check_positive_ordered	(	const char *	function,
		const char *	name,
		const Eigen::Matrix< T_y, Eigen::Dynamic, 1 > &	y
	)

Check if the specified vector contains non-negative values and is sorted into strictly increasing order.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Vector to test

Exceptions

<code>std::domain_error</code>

if the vector contains non-positive values, if the values are not ordered, if there are duplicated values, or if any element is NaN.

Definition at line 25 of file check_positive_ordered.hpp.

void stan::math::check_range ( const char *  function, const char *  name, int  max, int  index, int  nested_level, const char *  error_msg  )

inline

Check if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
max	Maximum size of the variable
index	Index to check
nested_level	Nested level (for error messages)
error_msg	Additional error message (for error messages)

Exceptions

<code>std::out_of_range</code>

if the index is not in range

Definition at line 24 of file check_range.hpp.

void stan::math::check_range ( const char *  function, const char *  name, int  max, int  index, const char *  error_msg  )

inline

Check if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
max	Maximum size of the variable
index	Index to check
error_msg	Additional error message (for error messages)

Exceptions

<code>std::out_of_range</code>

if the index is not in range

Definition at line 49 of file check_range.hpp.

void stan::math::check_range ( const char *  function, const char *  name, int  max, int  index  )

inline

Check if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
max	Maximum size of the variable
index	Index to check

Exceptions

<code>std::out_of_range</code>

if the index is not in range

Definition at line 69 of file check_range.hpp.

template<typename T_y , int R, int C>

void stan::math::check_row_index ( const char *  function, const char *  name, const Eigen::Matrix< T_y, R, C > &  y, size_t  i  )

inline

Check if the specified index is a valid row of the matrix This check is 1-indexed by default.

This behavior can be changed by setting stan::error_index::value.

Template Parameters

T	Scalar type
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	matrix to test
i	row index to check

Exceptions

<code>std::out_of_range</code>

if the index is out of range.

Definition at line 27 of file check_row_index.hpp.

template<typename T_prob >

void stan::math::check_simplex	(	const char *	function,
		const char *	name,
		const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &	theta
	)

Check if the specified vector is simplex.

To be a simplex, all values must be greater than or equal to 0 and the values must sum to 1. A valid simplex is one where the sum of hte elements is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accepts Eigen vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters

T_prob

Scalar type of the vector

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
theta	Vector to test.

Exceptions

<code>std::invalid_argument</code>	if theta is a 0-vector.
<code>std::domain_error</code>	if the vector is not a simplex or if any element is NaN.

Definition at line 34 of file check_simplex.hpp.

template<typename T_size1 , typename T_size2 >

void stan::math::check_size_match ( const char *  function, const char *  name_i, T_size1  i, const char *  name_j, T_size2  j  )

inline

Check if the provided sizes match.

Template Parameters

T_size1	Type of size 1
T_size2	Type of size 2

Parameters

function	Function name (for error messages)
name_i	Variable name 1 (for error messages)
i	Variable size 1
name_j	Variable name 2 (for error messages)
j	Variable size 2

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 24 of file check_size_match.hpp.

template<typename T_size1 , typename T_size2 >

void stan::math::check_size_match ( const char *  function, const char *  expr_i, const char *  name_i, T_size1  i, const char *  expr_j, const char *  name_j, T_size2  j  )

inline

Check if the provided sizes match.

Template Parameters

T_size1	Type of size 1
T_size2	Type of size 2

Parameters

function	Function name (for error messages)
expr_i	Expression for variable name 1 (for error messages)
name_i	Variable name 1 (for error messages)
i	Variable size 1
expr_j	Expression for variable name 2 (for error messages)
name_j	Variable name 2 (for error messages)
j	Variable size 2

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 50 of file check_size_match.hpp.

template<typename T_y >

void stan::math::check_square ( const char *  function, const char *  name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y  )

inline

Check if the specified matrix is square.

This check allows 0x0 matrices.

Template Parameters

T	Type of scalar

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>

if the matrix is not square

Definition at line 21 of file check_square.hpp.

template<typename T >

void stan::math::check_std_vector_index ( const char *  function, const char *  name, const std::vector< T > &  y, int  i  )

inline

Check if the specified index is valid in std vector This check is 1-indexed by default.

This behavior can be changed by setting stan::error_index::value.

Template Parameters

T	Scalar type

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	std::vector to test
i	Index

Exceptions

<code>std::out_of_range</code>

if the index is out of range.

Definition at line 26 of file check_std_vector_index.hpp.

template<typename T_y >

void stan::math::check_symmetric ( const char *  function, const char *  name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y  )

inline

Check if the specified matrix is symmetric.

The error message is either 0 or 1 indexed, specified by stan::error_index::value.

Template Parameters

T_y	Type of scalar

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
y	Matrix to test

Exceptions

<code>std::invalid_argument</code>	if the matrix is not square.
<code>std::domain_error</code>	if any element not on the main diagonal is NaN

Definition at line 31 of file check_symmetric.hpp.

template<typename T_prob >

void stan::math::check_unit_vector	(	const char *	function,
		const char *	name,
		const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &	theta
	)

Check if the specified vector is unit vector.

A valid unit vector is one where the square of the elements summed is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accepts Eigen vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters

T_prob

Scalar type of the vector

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
theta	Vector to test

Exceptions

<code>std::invalid_argument</code>	if theta is a 0-vector
<code>std::domain_error</code>	if the vector is not a unit vector or if any element is NaN

Definition at line 34 of file check_unit_vector.hpp.

template<typename T , int R, int C>

void stan::math::check_vector ( const char *  function, const char *  name, const Eigen::Matrix< T, R, C > &  x  )

inline

Check if the matrix is either a row vector or column vector.

This function checks the runtime size of the matrix to check whether it is a row or column vector.

Template Parameters

T	Scalar type of the matrix
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

function	Function name (for error messages)
name	Variable name (for error messages)
x	Matrix

Exceptions

<code>std::invalid_argument</code>

if x is not a row or column vector.

Definition at line 28 of file check_vector.hpp.

template<typename T_y , typename T_dof >

return_type_t<T_y, T_dof> stan::math::chi_square_lpdf ( const T_y &  y, const T_dof &  nu  )

inline

Definition at line 126 of file chi_square_lpdf.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::chol2inv

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

L

)

Returns the inverse of the matrix whose Cholesky factor is L.

Template Parameters

T	type of elements in the matrix

Parameters

L	Matrix that is a Cholesky factor.

Returns

The matrix inverse of L * L'

Exceptions

std::domain_error

If the input matrix is not square or lower triangular

Definition at line 24 of file chol2inv.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_corr_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	y,
		int	K
	)

Definition at line 15 of file cholesky_corr_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_corr_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	y,
		int	K,
		T &	lp
	)

Definition at line 48 of file cholesky_corr_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::cholesky_corr_free

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

x

)

Definition at line 14 of file cholesky_corr_free.hpp.

template<typename T , typename = require_floating_point_t<T>>

matrix_cl<T> stan::math::cholesky_decompose ( matrix_cl< T > &  A )

inline

Returns the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix on the OpenCL device.

The return value \(L\) will be a lower-traingular matrix such that the original matrix \(A\) is given by

\(A = L \times L^T\).

Parameters

A	Input square matrix

Returns

Square root of matrix.

Exceptions

std::domain_error

if m is not a symmetric matrix or if m is not positive definite (if m has more than 0 elements)

Definition at line 26 of file cholesky_decompose.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_decompose ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m )

inline

Return the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix.

The return value \(L\) will be a lower-triangular matrix such that the original matrix \(A\) is given by

\(A = L \times L^T\).

Template Parameters

T	type of elements in the matrix

Parameters

m	Symmetrix matrix.

Returns

Square root of matrix.

Note

Because OpenCL only works on doubles there are two cholesky_decompose functions. One that works on doubles and another that works on all other types (this one).

Exceptions

std::domain_error

if m is not a symmetric matrix or if m is not positive definite (if m has more than 0 elements)

Definition at line 32 of file cholesky_decompose.hpp.

template<>

Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_decompose ( const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  m )

inline

Return the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix.

The return value \(L\) will be a lower-triangular matrix such that the original matrix \(A\) is given by

\(A = L \times L^T\).

Parameters

m	Symmetrix matrix.

Returns

Square root of matrix.

Note

Because OpenCL only works on doubles there are two cholesky_decompose functions. One that works on doubles (this one) and another that works on all other types.

Exceptions

std::domain_error

if m is not a symmetric matrix or if m is not positive definite (if m has more than 0 elements)

Definition at line 57 of file cholesky_decompose.hpp.

Eigen::Matrix<var, -1, -1> stan::math::cholesky_decompose ( const Eigen::Matrix< var,-1,-1 > &  A )

inline

Reverse mode specialization of cholesky decomposition.

Internally calls Eigen::LLT rather than using stan::math::cholesky_decompose in order to use an inplace decomposition.

Note chainable stack varis are created below in Matrix<var, -1, -1>

Parameters

A

Matrix

Returns

L cholesky factor of A

Definition at line 376 of file cholesky_decompose.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_factor_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		int	M,
		int	N
	)

Return the Cholesky factor of the specified size read from the specified vector.

A total of (N choose 2) + N + (M - N) * N elements are required to read an M by N Cholesky factor.

Template Parameters

T	type of elements in the matrix

Parameters

x	Vector of unconstrained values
M	Number of rows
N	Number of columns

Returns

Cholesky factor

Definition at line 26 of file cholesky_factor_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_factor_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		int	M,
		int	N,
		T &	lp
	)

Return the Cholesky factor of the specified size read from the specified vector and increment the specified log probability reference with the log Jacobian adjustment of the transform.

A total of (N choose 2) + N + N * (M - N) free parameters are required to read an M by N Cholesky factor.

Template Parameters

T	type of elements in the matrix

Parameters

x	Vector of unconstrained values
M	Number of rows
N	Number of columns
lp	Log probability that is incremented with the log Jacobian

Returns

Cholesky factor

Definition at line 72 of file cholesky_factor_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::cholesky_factor_free

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

y

)

Return the unconstrained vector of parameters correspdonding to the specified Cholesky factor.

A Cholesky factor must be lower triangular and have positive diagonal elements.

Template Parameters

T	type of elements in the Cholesky factor

Parameters

y	Cholesky factor.

Returns

Unconstrained parameters for Cholesky factor.

Exceptions

std::domain_error

If the matrix is not a Cholesky factor.

Definition at line 23 of file cholesky_factor_free.hpp.

int stan::math::choose ( int  n, int  k  )

inline

Return the binomial coefficient for the specified integer arguments.

The binomial coefficient, \({n \choose k}\), read "n choose k", is defined for \(0 \leq k \leq n\) (otherwise return 0) by

\({n \choose k} = \frac{n!}{k! (n-k)!}\).

Parameters

n	total number of objects
k	number of objects chosen

Returns

n choose k or 0 iff k > n

Exceptions

std::domain_error

if either argument is negative or the result will not fit in an int type

Definition at line 28 of file choose.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::col ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m, size_t  j  )

inline

Return the specified column of the specified matrix using start-at-1 indexing.

This is equivalent to calling m.col(i - 1) and assigning the resulting template expression to a column vector.

Template Parameters

T	type of elements in the matrix

Parameters

m	Matrix.
j	Column index (count from 1).

Returns

Specified column of the matrix.

Exceptions

std::out_of_range

if j is out of range.

Definition at line 24 of file col.hpp.

template<typename T , int R, int C>

int stan::math::cols ( const Eigen::Matrix< T, R, C > &  m )

inline

Return the number of columns in the specified matrix, vector, or row vector.

Template Parameters

T	type of elements in the matrix
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]

m

Input matrix, vector, or row vector.

Returns

Number of columns.

Definition at line 20 of file cols.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2  )

inline

Definition at line 13 of file columns_dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename = require_any_var_t<T1, T2>>

Eigen::Matrix<return_type_t<T1, T2>, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< T1, R1, C1 > &  v1, const Eigen::Matrix< T2, R2, C2 > &  v2  )

inline

Definition at line 19 of file columns_dot_product.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<double, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Returns the dot product of the specified vectors.

Template Parameters

R1	number of rows, can be Eigen::Dynamic
C1	number of columns, can be Eigen::Dynamic
R2	number of rows, can be Eigen::Dynamic
C2	number of columns, can be Eigen::Dynamic

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 25 of file columns_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Definition at line 27 of file columns_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2  )

inline

Definition at line 41 of file columns_dot_product.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, 1, C> stan::math::columns_dot_self ( const Eigen::Matrix< fvar< T >, R, C > &  x )

inline

Definition at line 12 of file columns_dot_self.hpp.

template<int R, int C>

Eigen::Matrix<var, 1, C> stan::math::columns_dot_self ( const Eigen::Matrix< var, R, C > &  x )

inline

Returns the dot product of each column of a matrix with itself.

Parameters

x

Matrix.

Template Parameters

T	scalar type

Definition at line 19 of file columns_dot_self.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, 1, C> stan::math::columns_dot_self ( const Eigen::Matrix< T, R, C > &  x )

inline

Returns the dot product of each column of a matrix with itself.

Template Parameters

T	type of elements in the matrix
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

x

Matrix.

Definition at line 19 of file columns_dot_self.hpp.

template<typename T >

T stan::math::corr_constrain ( const T &  x )

inline

Return the result of transforming the specified scalar to have a valid correlation value between -1 and 1 (inclusive).

The transform used is the hyperbolic tangent function,

\(f(x) = \tanh x = \frac{\exp(2x) - 1}{\exp(2x) + 1}\).

Template Parameters

T	type of value

Parameters

[in]

x

value

Returns

tanh transform of value

Definition at line 25 of file corr_constrain.hpp.

template<typename T >

T stan::math::corr_constrain ( const T &  x, T &  lp  )

inline

Return the result of transforming the specified scalar to have a valid correlation value between -1 and 1 (inclusive).

The transform used is as specified for corr_constrain(T). The log absolute Jacobian determinant is

\(\log | \frac{d}{dx} \tanh x | = \log (1 - \tanh^2 x)\).

Template Parameters

T	Type of scalar.

Parameters

[in]	x	value
[in,out]	lp	log density accumulator

Definition at line 45 of file corr_constrain.hpp.

template<typename T >

T stan::math::corr_free ( const T &  y )

inline

Return the unconstrained scalar that when transformed to a valid correlation produces the specified value.

This function inverts the transform defined for corr_constrain(T), which is the inverse hyperbolic tangent,

\( f^{-1}(y) = \mbox{atanh}\, y = \frac{1}{2} \log \frac{y + 1}{y - 1}\).

Template Parameters

T	Type of correlation

Parameters

[in]

y

correlation

Returns

free scalar that transforms to the specified input

Definition at line 28 of file corr_free.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::corr_matrix_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 >>::type	k
	)

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length \({k \choose 2} = \frac{k(k-1)}{2}\). The values in the input vector represent unconstrained (partial) correlations among the dimensions.

The transform based on partial correlations is as specified in

• Lewandowski, Daniel, Dorota Kurowicka, and Harry Joe. 2009. Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis 100:1989–-2001.

The free vector entries are first constrained to be valid correlation values using corr_constrain(T).

Template Parameters

T	type of scalar

Parameters

x	Vector of unconstrained partial correlations.
k	Dimensionality of returned correlation matrix.

Exceptions

std::invalid_argument

if x is not a valid correlation matrix.

Definition at line 39 of file corr_matrix_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::corr_matrix_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 >>::type	k,
		T &	lp
	)

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length \({k \choose 2} = \frac{k(k-1)}{2}\). The values in the input vector represent unconstrained (partial) correlations among the dimensions.

The transform is as specified for corr_matrix_constrain(Matrix, size_t); the paper it cites also defines the Jacobians for correlation inputs, which are composed with the correlation constrained Jacobians defined in corr_constrain(T, double) for this function.

Template Parameters

T	type of scalar

Parameters

x	Vector of unconstrained partial correlations.
k	Dimensionality of returned correlation matrix.
lp	Log probability reference to increment.

Definition at line 76 of file corr_matrix_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::corr_matrix_free

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

y

)

Return the vector of unconstrained partial correlations that define the specified correlation matrix when transformed.

The constraining transform is defined as for corr_matrix_constrain(Matrix, size_t). The inverse transform in this function is simpler in that it only needs to compute the \(k \choose 2\) partial correlations and then free those.

Template Parameters

T	type of scalar

Parameters

y	The correlation matrix to free.

Returns

Vector of unconstrained values that produce the specified correlation matrix when transformed.

Exceptions

std::domain_error	if the correlation matrix has no elements or is not a square matrix.
std::runtime_error	if the correlation matrix cannot be factorized by factor_cov_matrix() or if the sds returned by factor_cov_matrix() on log scale are unconstrained.

Definition at line 34 of file corr_matrix_free.hpp.

template<typename T >

fvar<T> stan::math::cos ( const fvar< T > &  x )

inline

Definition at line 12 of file cos.hpp.

template<typename T >

apply_scalar_unary<cos_fun, T>::return_t stan::math::cos ( const T &  x )

inline

Vectorized version of cos().

Template Parameters

T	type of container

Parameters

x	angles in radians

Returns

Cosine of each value in x.

Definition at line 33 of file cos.hpp.

var stan::math::cos ( const var &  a )

inline

Return the cosine of a radian-scaled variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \cos x = - \sin x\).

\[ \mbox{cos}(x) = \begin{cases} \cos(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cos}(x)}{\partial x} = \begin{cases} -\sin(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable for radians of angle.

Returns

Cosine of variable.

Definition at line 46 of file cos.hpp.

template<typename T >

fvar<T> stan::math::cosh ( const fvar< T > &  x )

inline

Definition at line 12 of file cosh.hpp.

template<typename T >

apply_scalar_unary<cosh_fun, T>::return_t stan::math::cosh ( const T &  x )

inline

Vectorized version of cosh().

Template Parameters

T	type of container

Parameters

x	angles in radians

Returns

Hyberbolic cosine of x.

Definition at line 33 of file cosh.hpp.

var stan::math::cosh ( const var &  a )

inline

Return the hyperbolic cosine of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \cosh x = \sinh x\).

\[ \mbox{cosh}(x) = \begin{cases} \cosh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cosh}(x)}{\partial x} = \begin{cases} \sinh(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a

Variable.

Returns

Hyperbolic cosine of variable.

Definition at line 46 of file cosh.hpp.

template<typename T_x , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_exp_quad ( const std::vector< T_x > &  x, const T_sigma &  sigma, const T_l &  length_scale  )

inline

Deprecated:

use gp_exp_quad_cov

Definition at line 20 of file cov_exp_quad.hpp.

template<typename T_x , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_exp_quad ( const std::vector< T_x > &  x, const T_sigma &  sigma, const std::vector< T_l > &  length_scale  )

inline

Deprecated:

use gp_exp_quad_cov

Definition at line 31 of file cov_exp_quad.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_exp_quad ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_sigma &  sigma, const T_l &  length_scale  )

inline

Deprecated:

use gp_exp_quad_cov

Definition at line 42 of file cov_exp_quad.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_exp_quad ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_sigma &  sigma, const std::vector< T_l > &  length_scale  )

inline

Deprecated:

use gp_exp_quad_cov

Definition at line 53 of file cov_exp_quad.hpp.

template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>

Eigen::Matrix<var, -1, -1> stan::math::cov_exp_quad ( const std::vector< T_x > &  x, const var &  sigma, const var &  l  )

inline

Deprecated:

use gp_exp_quad_cov_vari

Definition at line 154 of file cov_exp_quad.hpp.

template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>

Eigen::Matrix<var, -1, -1> stan::math::cov_exp_quad ( const std::vector< T_x > &  x, double  sigma, const var &  l  )

inline

Deprecated:

use gp_exp_quad_cov_vari

Definition at line 164 of file cov_exp_quad.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 >>::type	K
	)

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

See cov_matrix_free() for the inverse transform.

Template Parameters

T	type of elements in the vector

Parameters

x	The vector to convert to a covariance matrix.
K	The number of rows and columns of the resulting covariance matrix.

Exceptions

std::invalid_argument

if (x.size() != K + (K choose 2)).

Definition at line 28 of file cov_matrix_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic >>::type	K,
		T &	lp
	)

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

See cov_matrix_free() for the inverse transform.

Template Parameters

T	type of elements in the vector

Parameters

x	The vector to convert to a covariance matrix.
K	The dimensions of the resulting covariance matrix.
lp	Reference

Exceptions

std::domain_error

if (x.size() != K + (K choose 2)).

Definition at line 66 of file cov_matrix_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain_lkj	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		size_t	k
	)

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values.

The input vector must be of length \(k \choose 2 + k\). The first \(k \choose 2\) values in the input represent unconstrained (partial) correlations and the last \(k\) are unconstrained standard deviations of the dimensions.

The transform scales the correlation matrix transform defined in corr_matrix_constrain(Matrix, size_t) with the constrained deviations.

Template Parameters

T	type of elements in the vector

Parameters

x	Input vector of unconstrained partial correlations and standard deviations.
k	Dimensionality of returned covariance matrix.

Returns

Covariance matrix derived from the unconstrained partial correlations and deviations.

Definition at line 32 of file cov_matrix_constrain_lkj.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain_lkj	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		size_t	k,
		T &	lp
	)

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values and increment the specified log probability reference with the log absolute Jacobian determinant.

The transform is defined as for cov_matrix_constrain(Matrix, size_t).

The log absolute Jacobian determinant is derived by composing the log absolute Jacobian determinant for the underlying correlation matrix as defined in cov_matrix_constrain(Matrix, size_t, T&) with the Jacobian of the transform of the correlation matrix into a covariance matrix by scaling by standard deviations.

Template Parameters

T	type of elements in the vector

Parameters

x	Input vector of unconstrained partial correlations and standard deviations.
k	Dimensionality of returned covariance matrix.
lp	Log probability reference to increment.

Returns

Covariance matrix derived from the unconstrained partial correlations and deviations.

Definition at line 72 of file cov_matrix_constrain_lkj.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::cov_matrix_free

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

y

)

The covariance matrix derived from the symmetric view of the lower-triangular view of the K by K specified matrix is freed to return a vector of size K + (K choose 2).

This is the inverse of the cov_matrix_constrain() function so that for any finite vector x of size K

• (K choose 2),

x == cov_matrix_free(cov_matrix_constrain(x, K)).

In order for this round-trip to work (and really for this function to work), the symmetric view of its lower-triangular view must be positive definite.

Template Parameters

T	type of elements in the matrix

Parameters

y	Matrix of dimensions K by K such that he symmetric view of the lower-triangular view is positive definite.

Returns

Vector of size K plus (K choose 2) in (-inf, inf) that produces

Exceptions

std::domain_error

if y is not square, has zero dimensionality, or has a non-positive diagonal element.

Definition at line 36 of file cov_matrix_free.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::cov_matrix_free_lkj

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

y

)

Return the vector of unconstrained partial correlations and deviations that transform to the specified covariance matrix.

The constraining transform is defined as for cov_matrix_constrain(Matrix, size_t). The inverse first factors out the deviations, then applies the freeing transform of corr_matrix_free(Matrix&).

Template Parameters

T	type of elements in the matrix

Parameters

y	Covariance matrix to free.

Returns

Vector of unconstrained values that transforms to the specified covariance matrix.

Exceptions

std::domain_error	if the correlation matrix has no elements or is not a square matrix.
std::runtime_error	if the correlation matrix cannot be factorized by factor_cov_matrix()

Definition at line 30 of file cov_matrix_free_lkj.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, C, C> stan::math::crossprod ( const Eigen::Matrix< fvar< T >, R, C > &  m )

inline

Definition at line 12 of file crossprod.hpp.

matrix_v stan::math::crossprod ( const matrix_v &  M )

inline

Returns the result of pre-multiplying a matrix by its own transpose.

Parameters

M	Matrix to multiply.

Returns

Transpose of M times M

Definition at line 17 of file crossprod.hpp.

matrix_d stan::math::crossprod ( const matrix_d &  M )

inline

Returns the result of pre-multiplying a matrix by its own transpose.

Parameters

M	Matrix to multiply.

Returns

Transpose of M times M

Definition at line 17 of file crossprod.hpp.

template<typename T1 , typename T2 >

Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, 1> stan::math::csr_matrix_times_vector ( int  m, int  n, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &  w, const std::vector< int > &  v, const std::vector< int > &  u, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &  b  )

inline

Definition at line 74 of file csr_matrix_times_vector.hpp.

template<typename T >

std::vector<T> stan::math::cumulative_sum ( const std::vector< T > &  x )

inline

Return the cumulative sum of the specified vector.

The cumulative sum of a vector of values

 is the

\code x[0], x[1] + x[2], ..., x[1] + , ..., + x[x.size()-1] 

Template Parameters

T	type of elements in the vector

Parameters

x	Vector of values.

Returns

Cumulative sum of values.

Definition at line 24 of file cumulative_sum.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::cumulative_sum ( const Eigen::Matrix< T, R, C > &  m )

inline

Return the cumulative sum of the specified matrix.

The cumulative sum is of the same type as the input and has values defined by

x(0), x(1) + x(2), ..., x(1) + , ..., + x(x.size()-1) 

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	Matrix of values.

Returns

Cumulative sum of values.

Definition at line 49 of file cumulative_sum.hpp.

void stan::math::cvodes_set_options ( void *  cvodes_mem, double  rel_tol, double  abs_tol, long int  max_num_steps  )

inline

Definition at line 20 of file cvodes_utils.hpp.

void stan::math::cvodes_silent_err_handler ( int  error_code, const char *  module, const char *  function, char *  msg, void *  eh_data  )

inline

Definition at line 15 of file cvodes_utils.hpp.

template<typename T , typename F >

void stan::math::derivative	(	const F &	f,
		const T &	x,
		T &	fx,
		T &	dfx_dx
	)

Return the derivative of the specified univariate function at the specified argument.

Template Parameters

T	Argument type
F	Function type

Parameters

[in]	f	Function
[in]	x	Argument
[out]	fx	Value of function applied to argument
[out]	dfx_dx	Value of derivative

Definition at line 24 of file derivative.hpp.

template<typename T , int R, int C>

fvar<T> stan::math::determinant ( const Eigen::Matrix< fvar< T >, R, C > &  m )

inline

Definition at line 12 of file determinant.hpp.

template<typename T , int R, int C>

T stan::math::determinant ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the determinant of the specified square matrix.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	Specified matrix.

Returns

Determinant of the matrix.

Exceptions

std::domain_error

if matrix is not square.

Definition at line 22 of file determinant.hpp.

template<int R, int C>

var stan::math::determinant ( const Eigen::Matrix< var, R, C > &  m )

inline

Definition at line 48 of file determinant.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::diag_matrix ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v )

inline

Return a square diagonal matrix with the specified vector of coefficients as the diagonal values.

Template Parameters

T	type of elements in the vector

Parameters

[in]

v

Specified vector.

Returns

Diagonal matrix with vector as diagonal values.

Definition at line 18 of file diag_matrix.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C1> stan::math::diag_post_multiply	(	const Eigen::Matrix< T1, R1, C1 > &	m1,
		const Eigen::Matrix< T2, R2, C2 > &	m2
	)

Definition at line 11 of file diag_post_multiply.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R2, C2> stan::math::diag_pre_multiply	(	const Eigen::Matrix< T1, R1, C1 > &	m1,
		const Eigen::Matrix< T2, R2, C2 > &	m2
	)

Definition at line 11 of file diag_pre_multiply.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::diagonal ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m )

inline

Return a column vector of the diagonal elements of the specified matrix.

The matrix is not required to be square.

Template Parameters

T	type of elements in the matrix

Parameters

m	Specified matrix.

Returns

Diagonal of the matrix.

Definition at line 18 of file diagonal.hpp.

var stan::math::digamma ( const var &  a )

inline

Definition at line 20 of file digamma.hpp.

template<typename T >

fvar<T> stan::math::digamma ( const fvar< T > &  x )

inline

Return the derivative of the log gamma function at the specified argument.

Template Parameters

T	scalar type of autodiff variable

Parameters

[in]

x

argument

Returns

derivative of the log gamma function at the specified argument

Definition at line 23 of file digamma.hpp.

template<typename T >

apply_scalar_unary<digamma_fun, T>::return_t stan::math::digamma ( const T &  x )

inline

Vectorized version of digamma().

Template Parameters

T	type of container

Parameters

x

container

Returns

Digamma function applied to each value in x.

Exceptions

std::domain_error

if any value is a negative integer or 0

Definition at line 34 of file digamma.hpp.

double stan::math::digamma ( double  x )

inline

Return the derivative of the log gamma function at the specified value.

\[ \mbox{digamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{digamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Psi(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Psi(x)=\frac{\Gamma'(x)}{\Gamma(x)} \]

\[ \frac{\partial \, \Psi(x)}{\partial x} = \frac{\Gamma''(x)\Gamma(x)-(\Gamma'(x))^2}{\Gamma^2(x)} \]

The design follows the standard C++ library in returning NaN rather than throwing exceptions.

Parameters

[in]

x

argument

Returns

derivative of log gamma function at argument

Definition at line 46 of file digamma.hpp.

template<typename T >

void stan::math::dims ( const T &  x, std::vector< int > &  result  )

inline

Definition at line 11 of file dims.hpp.

template<typename T , int R, int C>

void stan::math::dims ( const Eigen::Matrix< T, R, C > &  x, std::vector< int > &  result  )

inline

Definition at line 15 of file dims.hpp.

template<typename T >

void stan::math::dims ( const std::vector< T > &  x, std::vector< int > &  result  )

inline

Definition at line 20 of file dims.hpp.

template<typename T >

std::vector<int> stan::math::dims ( const T &  x )

inline

Definition at line 28 of file dims.hpp.

template<typename T_prob , typename T_prior_size >

return_type_t<T_prob, T_prior_size> stan::math::dirichlet_lpdf	(	const T_prob &	theta,
		const T_prior_size &	alpha
	)

Definition at line 116 of file dirichlet_lpdf.hpp.

template<typename T_prob , typename T_prior_size >

return_type_t<T_prob, T_prior_size> stan::math::dirichlet_lpmf	(	const T_prob &	theta,
		const T_prior_size &	alpha
	)

Definition at line 20 of file dirichlet_lpmf.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::distance ( const T1 &  x1, const T2 &  x2  )

inline

Returns the distance between two scalars.

Parameters

x1	First scalar.
x2	Second scalar.

Returns

Distance between two scalars

Exceptions

std::domain_error

If the arguments are not finite.

Definition at line 20 of file distance.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>

return_type_t<T1, T2> stan::math::distance ( const Eigen::Matrix< T1, R1, C1 > &  v1, const Eigen::Matrix< T2, R2, C2 > &  v2  )

inline

Returns the distance between the specified vectors.

Template Parameters

T1	type of elements in first vector
T2	type of elements in second vector

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 24 of file distance.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::divide ( const Eigen::Matrix< fvar< T >, R, C > &  v, const fvar< T > &  c  )

inline

Definition at line 11 of file divide.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::divide ( const T1 &  x, const T2 &  y  )

inline

Return the division of the first scalar by the second scalar.

Parameters

[in]	x	Specified vector.
[in]	y	Specified scalar.

Returns

Vector divided by the scalar.

Definition at line 20 of file divide.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::divide ( const Eigen::Matrix< fvar< T >, R, C > &  v, double  c  )

inline

Definition at line 20 of file divide.hpp.

int stan::math::divide ( int  x, int  y  )

inline

Definition at line 24 of file divide.hpp.

template<int R, int C, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::divide ( const Eigen::Matrix< T1, R, C > &  m, T2  c  )

inline

Return matrix divided by scalar.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic
T1	type of elements in the matrix
T2	type of scalar

Parameters

[in]	m	specified matrix
[in]	c	specified scalar

Returns

matrix divided by the scalar

Definition at line 24 of file divide.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::divide ( const Eigen::Matrix< double, R, C > &  v, const fvar< T > &  c  )

inline

Definition at line 29 of file divide.hpp.

template<int R, int C>

Eigen::Matrix<var, R, C> stan::math::divide ( const Eigen::Matrix< double, R, C > &  m, const var &  c  )

inline

Return matrix divided by scalar.

Template Parameters

R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	m	specified matrix
[in]	c	specified scalar

Returns

matrix divided by the scalar

Definition at line 123 of file divide.hpp.

template<int R, int C>

Eigen::Matrix<var, R, C> stan::math::divide ( const Eigen::Matrix< var, R, C > &  m, const double &  c  )

inline

Return matrix divided by scalar.

Template Parameters

R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	m	specified matrix
[in]	c	specified scalar

Returns

matrix divided by the scalar

Definition at line 142 of file divide.hpp.

template<int R, int C>

Eigen::Matrix<var, R, C> stan::math::divide ( const Eigen::Matrix< var, R, C > &  m, const var &  c  )

inline

Return matrix divided by scalar.

Template Parameters

R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	m	specified matrix
[in]	c	specified scalar

Returns

matrix divided by the scalar

Definition at line 161 of file divide.hpp.

template<typename T_x , typename T_v >

std::vector< Eigen::Matrix<return_type_t<T_x, T_v, double>, Eigen::Dynamic, 1> > stan::math::divide_columns ( const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &  x, const std::vector< T_v > &  vec  )

inline

Takes Stan data type vector[n] x[D] and divides column vector in x element-wise by the values in vec.

Template Parameters

T_x	Type of dividend
T_v	Scalar type of divisor

Parameters

x	std::vector of matrices
vec	std::vector of divisors

Exceptions

std::invalid

argument if D != length of vector

Definition at line 26 of file divide_columns.hpp.

template<typename T_shape >

return_type_t<double, T_shape> stan::math::do_lkj_constant	(	const T_shape &	eta,
		const unsigned int &	K
	)

Definition at line 13 of file lkj_corr_lpdf.hpp.

template<typename T >

void stan::math::domain_error ( const char *  function, const char *  name, const T &  y, const char *  msg1, const char *  msg2  )

inline

Deprecated:

use throw_domain_error

Definition at line 14 of file domain_error.hpp.

template<typename T >

void stan::math::domain_error ( const char *  function, const char *  name, const T &  y, const char *  msg1  )

inline

Deprecated:

use throw_domain_error

Definition at line 23 of file domain_error.hpp.

template<typename T >

void stan::math::domain_error_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg1, const char *  msg2  )

inline

Deprecated:

use throw_domain_error_vec

Definition at line 13 of file domain_error_vec.hpp.

template<typename T >

void stan::math::domain_error_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg1  )

inline

Deprecated:

use throw_domain_error_vec

Definition at line 22 of file domain_error_vec.hpp.

double stan::math::dot ( const std::vector< double > &  x, const std::vector< double > &  y  )

inline

Definition at line 11 of file dot.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2  )

inline

Definition at line 13 of file dot_product.hpp.

template<int R1, int C1, int R2, int C2>

double stan::math::dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Returns the dot product of the specified vectors.

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 21 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Definition at line 27 of file dot_product.hpp.

double stan::math::dot_product ( const double *  v1, const double *  v2, size_t  length  )

inline

Returns the dot product of the specified arrays of doubles.

Parameters

v1	First array.
v2	Second array.
length	Length of both arrays.

Definition at line 36 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2  )

inline

Definition at line 41 of file dot_product.hpp.

double stan::math::dot_product ( const std::vector< double > &  v1, const std::vector< double > &  v2  )

inline

Returns the dot product of the specified arrays of doubles.

Parameters

v1	First array.
v2	Second array.

Exceptions

std::domain_error

if the vectors are not the same size.

Definition at line 51 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2, size_type &  length  )

inline

Definition at line 55 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2, size_type &  length  )

inline

Definition at line 69 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2, size_type &  length  )

inline

Definition at line 83 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const std::vector< fvar< T > > &  v1, const std::vector< fvar< T > > &  v2  )

inline

Definition at line 97 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const std::vector< double > &  v1, const std::vector< fvar< T > > &  v2  )

inline

Definition at line 108 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const std::vector< fvar< T > > &  v1, const std::vector< double > &  v2  )

inline

Definition at line 119 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const std::vector< fvar< T > > &  v1, const std::vector< fvar< T > > &  v2, size_type &  length  )

inline

Definition at line 130 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const std::vector< double > &  v1, const std::vector< fvar< T > > &  v2, size_type &  length  )

inline

Definition at line 140 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const std::vector< fvar< T > > &  v1, const std::vector< double > &  v2, size_type &  length  )

inline

Definition at line 150 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const fvar< T > *  v1, const fvar< T > *  v2, size_type  length  )

inline

Return dot product of specified pointers up to specified length.

Template Parameters

T	type of scalar within fvar

Parameters

v1	pointer to first sequence
v2	pointer second sequence
length	number of elements to multiply from each sequence

Returns

dot product of sequences up to length

Definition at line 169 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const double *  v1, const fvar< T > *  v2, size_type  length  )

inline

Return dot product of specified pointers up to specified length.

Template Parameters

T	type of scalar within fvar

Parameters

v1	pointer to first sequence
v2	pointer second sequence
length	number of elements to multiply from each sequence

Returns

dot product of sequences up to length

Definition at line 187 of file dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename = require_any_var_t<T1, T2>>

return_type_t<T1, T2> stan::math::dot_product ( const Eigen::Matrix< T1, R1, C1 > &  v1, const Eigen::Matrix< T2, R2, C2 > &  v2  )

inline

Returns the dot product.

Parameters

[in]	v1	First column vector.
[in]	v2	Second column vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

if length of v1 is not equal to length of v2.

Definition at line 193 of file dot_product.hpp.

template<typename T >

fvar<T> stan::math::dot_product ( const fvar< T > *  v1, const double *  v2, size_type  length  )

inline

Return dot product of specified pointers up to specified length.

Template Parameters

T	type of scalar within fvar

Parameters

v1	pointer to first sequence
v2	pointer second sequence
length	number of elements to multiply from each sequence

Returns

dot product of sequences up to length

Definition at line 205 of file dot_product.hpp.

template<typename T1 , typename T2 , typename = require_any_var_t<T1, T2>>

return_type_t<T1, T2> stan::math::dot_product ( const T1 *  v1, const T2 *  v2, size_t  length  )

inline

Returns the dot product.

Parameters

[in]	v1	First array.
[in]	v2	Second array.
[in]	length	Length of both arrays.

Returns

Dot product of the arrays.

Definition at line 209 of file dot_product.hpp.

template<typename T1 , typename T2 , typename = require_any_var_t<T1, T2>>

return_type_t<T1, T2> stan::math::dot_product ( const std::vector< T1 > &  v1, const std::vector< T2 > &  v2  )

inline

Returns the dot product.

Parameters

[in]	v1	First vector.
[in]	v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

if sizes of v1 and v2 do not match.

Definition at line 223 of file dot_product.hpp.

double stan::math::dot_self ( const std::vector< double > &  x )

inline

Definition at line 11 of file dot_self.hpp.

template<typename T , int R, int C>

fvar<T> stan::math::dot_self ( const Eigen::Matrix< fvar< T >, R, C > &  v )

inline

Definition at line 13 of file dot_self.hpp.

template<int R, int C>

double stan::math::dot_self ( const Eigen::Matrix< double, R, C > &  v )

inline

Returns the dot product of the specified vector with itself.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

v

Vector.

Exceptions

std::domain_error

If v is not vector dimensioned.

Definition at line 19 of file dot_self.hpp.

template<int R, int C>

var stan::math::dot_self ( const Eigen::Matrix< var, R, C > &  v )

inline

Returns the dot product of a vector with itself.

Parameters

[in]

v

Vector.

Returns

Dot product of the vector with itself.

Template Parameters

R	number of rows or Eigen::Dynamic for dynamic; one of R or C must be 1
C	number of rows or Eigen::Dyanmic for dynamic; one of R or C must be 1

Definition at line 67 of file dot_self.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::double_exponential_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 112 of file double_exponential_lpdf.hpp.

double stan::math::e ( )

inline

Return the base of the natural logarithm.

Returns

Base of natural logarithm.

Definition at line 85 of file constants.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::eigenvalues_sym

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Return the eigenvalues of the specified symmetric matrix in descending order of magnitude.

This function is more efficient than the general eigenvalues function for symmetric matrices.

See eigen_decompose() for more information.

Template Parameters

T	type of elements in the matrix

Parameters

m	Specified matrix.

Returns

Eigenvalues of matrix.

Definition at line 22 of file eigenvalues_sym.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::eigenvectors_sym

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Definition at line 11 of file eigenvectors_sym.hpp.

template<typename T_a , typename T_b , typename = require_all_valid_expressions_t<T_a, T_b>>

elewise_division_<as_operation_cl_t<T_a>, as_operation_cl_t<T_b> > stan::math::elewise_division ( T_a &&  a, T_b &&  b  )

inline

Element-wise division of two kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Element-wise division of given expressions

Definition at line 291 of file binary_operation.hpp.

template<typename T_a , typename T_b >

elewise_multiplication_<as_operation_cl_t<T_a>, as_operation_cl_t<T_b> > stan::math::elewise_multiplication ( T_a &&  a, T_b &&  b  )

inline

Element-wise multiplication of two kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Element-wise multiplication of given expressions

Definition at line 197 of file binary_operation.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::elt_divide	(	const Eigen::Matrix< T1, R, C > &	m1,
		const Eigen::Matrix< T2, R, C > &	m2
	)

Return the elementwise division of the specified matrices.

Template Parameters

T1	type of elements in first matrix
T2	type of elements in second matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m1	First matrix
m2	Second matrix

Returns

Elementwise division of matrices.

Definition at line 23 of file elt_divide.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::elt_divide	(	const Eigen::Matrix< T1, R, C > &	m,
		T2	s
	)

Return the elementwise division of the specified matrix by the specified scalar.

Template Parameters

T1	type of elements in the matrix
T2	type of the scalar
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	matrix
s	scalar

Returns

Elementwise division of a scalar by matrix.

Definition at line 44 of file elt_divide.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::elt_divide	(	T1	s,
		const Eigen::Matrix< T2, R, C > &	m
	)

Return the elementwise division of the specified scalar by the specified matrix.

Template Parameters

T1	type of the scalar
T2	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

s	scalar
m	matrix

Returns

Elementwise division of a scalar by matrix.

Definition at line 63 of file elt_divide.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::elt_multiply	(	const Eigen::Matrix< T1, R, C > &	m1,
		const Eigen::Matrix< T2, R, C > &	m2
	)

Return the elementwise multiplication of the specified matrices.

Template Parameters

T1	type of elements in first matrix
T2	type of elements in second matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m1	First matrix
m2	Second matrix

Returns

Elementwise product of matrices.

Definition at line 24 of file elt_multiply.hpp.

static bool stan::math::empty_nested ( )

inlinestatic

Return true if there is no nested autodiff being executed.

Definition at line 12 of file empty_nested.hpp.

template<typename T >

fvar<T> stan::math::erf ( const fvar< T > &  x )

inline

Definition at line 15 of file erf.hpp.

double stan::math::erf ( double  x )

inline

Return the error function of the specified value.

\[ \mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt \]

Parameters

[in]

x

Argument.

Returns

Error function of the argument.

Definition at line 20 of file erf.hpp.

double stan::math::erf ( int  x )

inline

Return the error function of the specified argument.

This version is required to disambiguate erf(int).

Parameters

[in]

x

Argument.

Returns

Error function of the argument.

Definition at line 29 of file erf.hpp.

template<typename T >

apply_scalar_unary<erf_fun, T>::return_t stan::math::erf ( const T &  x )

inline

Vectorized version of erf().

Template Parameters

T	type of container

Parameters

x

container

Returns

Error function applied to each value in x.

Definition at line 32 of file erf.hpp.

var stan::math::erf ( const var &  a )

inline

The error function for variables (C99).

The derivative is

\(\frac{d}{dx} \mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \exp(-x^2)\).

\[ \mbox{erf}(x) = \begin{cases} \operatorname{erf}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{erf}(x)}{\partial x} = \begin{cases} \frac{\partial\, \operatorname{erf}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \]

\[ \frac{\partial \, \operatorname{erf}(x)}{\partial x} = \frac{2}{\sqrt{\pi}} e^{-x^2} \]

Parameters

a	The variable.

Returns

Error function applied to the variable.

Definition at line 58 of file erf.hpp.

template<typename T >

fvar<T> stan::math::erfc ( const fvar< T > &  x )

inline

Definition at line 15 of file erfc.hpp.

double stan::math::erfc ( double  x )

inline

Return the complementary error function of the specified value.

\[ \mbox{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt \]

Parameters

[in]

x

Argument.

Returns

Complementary error function of the argument.

Definition at line 20 of file erfc.hpp.

double stan::math::erfc ( int  x )

inline

Return the error function of the specified argument.

This version is required to disambiguate erfc(int).

Parameters

[in]

x

Argument.

Returns

Complementary error function value of the argument.

Definition at line 29 of file erfc.hpp.

template<typename T >

apply_scalar_unary<erfc_fun, T>::return_t stan::math::erfc ( const T &  x )

inline

Vectorized version of erfc().

Template Parameters

T	type of container

Parameters

x

container

Returns

Complementary error function applied to each value in x.

Definition at line 32 of file erfc.hpp.

var stan::math::erfc ( const var &  a )

inline

The complementary error function for variables (C99).

The derivative is

\(\frac{d}{dx} \mbox{erfc}(x) = - \frac{2}{\sqrt{\pi}} \exp(-x^2)\).

\[ \mbox{erfc}(x) = \begin{cases} \operatorname{erfc}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{erfc}(x)}{\partial x} = \begin{cases} \frac{\partial\, \operatorname{erfc}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_x^\infty e^{-t^2}dt \]

\[ \frac{\partial \, \operatorname{erfc}(x)}{\partial x} = -\frac{2}{\sqrt{\pi}} e^{-x^2} \]

Parameters

a	The variable.

Returns

Complementary error function applied to the variable.

Definition at line 59 of file erfc.hpp.

template<typename T >

fvar<T> stan::math::exp ( const fvar< T > &  x )

inline

Definition at line 11 of file exp.hpp.

double stan::math::exp ( int  x )

inline

Return the natural exponential of the specified argument.

This version is required to disambiguate exp(int).

Parameters

[in]

x

Argument.

Returns

Natural exponential of argument.

Definition at line 17 of file exp.hpp.

template<typename T >

apply_scalar_unary<exp_fun, T>::return_t stan::math::exp ( const T &  x )

inline

Return the elementwise exponentiation of the specified argument, which may be a scalar or any Stan container of numeric scalars.

The return type is the same as the argument type.

Template Parameters

T	type of container

Parameters

[in]

x

container

Returns

Elementwise application of exponentiation to the argument.

Definition at line 40 of file exp.hpp.

var stan::math::exp ( const var &  a )

inline

Return the exponentiation of the specified variable (cmath).

\[ \mbox{exp}(x) = \begin{cases} e^x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{exp}(x)}{\partial x} = \begin{cases} e^x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable to exponentiate.

Returns

Exponentiated variable.

Definition at line 41 of file exp.hpp.

template<typename T >

fvar<T> stan::math::exp2 ( const fvar< T > &  x )

inline

Definition at line 14 of file exp2.hpp.

template<typename T , typename = require_arithmetic_t<T>>

double stan::math::exp2 ( T  y )

inline

Return the exponent base 2 of the specified argument (C99, C++11).

The exponent base 2 function is defined by

exp2(y) = pow(2.0, y).

Parameters

y

argument.

Returns

exponent base 2 of argument.

Definition at line 22 of file exp2.hpp.

template<typename T , typename = require_vector_like_t<T>>

apply_scalar_unary<exp2_fun, T>::return_t stan::math::exp2 ( const T &  x )

inline

Return the elementwise application of exp2() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise exp2 of members of container.

Definition at line 39 of file exp2.hpp.

var stan::math::exp2 ( const var &  a )

inline

Exponentiation base 2 function for variables (C99).

The derivative is

\(\frac{d}{dx} 2^x = (\log 2) 2^x\).

\[ \mbox{exp2}(x) = \begin{cases} 2^x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{exp2}(x)}{\partial x} = \begin{cases} 2^x\ln2 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	The variable.

Returns

Two to the power of the specified variable.

Definition at line 46 of file exp2.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_inv_scale &	lambda
	)

Definition at line 18 of file exp_mod_normal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_inv_scale &	lambda
	)

Definition at line 18 of file exp_mod_normal_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_inv_scale &	lambda
	)

Definition at line 18 of file exp_mod_normal_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_inv_scale &	lambda
	)

Definition at line 17 of file exp_mod_normal_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >

return_type_t<T_y, T_loc, T_scale, T_inv_scale> stan::math::exp_mod_normal_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma, const T_inv_scale &  lambda  )

inline

Definition at line 106 of file exp_mod_normal_lpdf.hpp.

template<typename T >

fvar<T> stan::math::expm1 ( const fvar< T > &  x )

inline

Definition at line 13 of file expm1.hpp.

double stan::math::expm1 ( double  x )

inline

Return the natural exponentiation of x minus one.

Returns infinity for infinity argument and -infinity for -infinity argument.

Parameters

[in]

x

Argument.

Returns

Natural exponentiation of argument minus one.

Definition at line 18 of file expm1.hpp.

double stan::math::expm1 ( int  x )

inline

Integer version of expm1.

Parameters

[in]

x

Argument.

Returns

Natural exponentiation of argument minus one.

Definition at line 26 of file expm1.hpp.

template<typename T >

apply_scalar_unary<expm1_fun, T>::return_t stan::math::expm1 ( const T &  x )

inline

Vectorized version of expm1().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural exponential of each value in x minus one.

Definition at line 32 of file expm1.hpp.

var stan::math::expm1 ( const var &  a )

inline

The exponentiation of the specified variable minus 1 (C99).

The derivative is given by

\(\frac{d}{dx} \exp(a) - 1 = \exp(a)\).

\[ \mbox{expm1}(x) = \begin{cases} e^x-1 & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{expm1}(x)}{\partial x} = \begin{cases} e^x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	The variable.

Returns

Two to the power of the specified variable.

Definition at line 46 of file expm1.hpp.

template<typename T_y , typename T_inv_scale >

return_type_t<T_y, T_inv_scale> stan::math::exponential_lccdf	(	const T_y &	y,
		const T_inv_scale &	beta
	)

Definition at line 13 of file exponential_lccdf.hpp.

template<typename T_y , typename T_inv_scale >

return_type_t<T_y, T_inv_scale> stan::math::exponential_lcdf	(	const T_y &	y,
		const T_inv_scale &	beta
	)

Definition at line 14 of file exponential_lcdf.hpp.

template<typename T_y , typename T_inv_scale >

return_type_t<T_y, T_inv_scale> stan::math::exponential_lpdf ( const T_y &  y, const T_inv_scale &  beta  )

inline

Definition at line 94 of file exponential_lpdf.hpp.

template<typename T >

T stan::math::F32	(	const T &	a1,
		const T &	a2,
		const T &	a3,
		const T &	b1,
		const T &	b2,
		const T &	z,
		double	precision = 1e-6,
		int	max_steps = 1e5
	)

Hypergeometric function (3F2).

Function reference: http://dlmf.nist.gov/16.2

\[ _3F_2 \left( \begin{matrix}a_1 a_2 a3 \\ b_1 b_2\end{matrix}; z \right) = \sum_k=0^\infty \frac{(a_1)_k(a_2)_k(a_3)_k}{(b_1)_k(b_2)_k}\frac{z^k}{k!} \]

Where _k$ is an upper shifted factorial.

Calculate the hypergeometric function (3F2) as the power series directly to within precision or until max_steps terms.

This function does not have a closed form but will converge if:

• |z| is less than 1
• |z| is equal to one and b1 + b2 < a1 + a2 + a3 This function is a rational polynomial if
• a1, a2, or a3 is a non-positive integer This function can be treated as a rational polynomial if
• b1 or b2 is a non-positive integer and the series is terminated prior to the final term.

Template Parameters

T	type of arguments and result

Parameters

[in]	a1	a1 (always called with 1 from beta binomial cdfs)
[in]	a2	a2 (always called with a2 > 1)
[in]	a3	a3 (always called with int a3 <= 0)
[in]	b1	b1 (always called with int b1 < |a3|)
[in]	b2	b2 (always <= 1)
[in]	z	z (is always called with 1 from beta binomial cdfs)
[in]	precision	precision of the infinite sum. defaults to 1e-6
[in]	max_steps	number of steps to take. defaults to 1e5

Definition at line 50 of file F32.hpp.

template<typename T >

fvar<T> stan::math::fabs ( const fvar< T > &  x )

inline

Definition at line 15 of file fabs.hpp.

template<typename T >

apply_scalar_unary<fabs_fun, T>::return_t stan::math::fabs ( const T &  x )

inline

Vectorized version of fabs().

Template Parameters

T	type of container

Parameters

x

container

Returns

Absolute value of each value in x.

Definition at line 33 of file fabs.hpp.

var stan::math::fabs ( const var &  a )

inline

Return the absolute value of the variable (cmath).

Choosing an arbitrary value at the non-differentiable point 0,

\(\frac{d}{dx}|x| = \mbox{sgn}(x)\).

where \(\mbox{sgn}(x)\) is the signum function, taking values -1 if \(x < 0\), 0 if \(x == 0\), and 1 if \(x == 1\).

The function abs() provides the same behavior, with abs() defined in stdlib.h and fabs() defined in cmath. The derivative is 0 if the input is 0.

Returns std::numeric_limits<double>::quiet_NaN() for NaN inputs.

\[ \mbox{fabs}(x) = \begin{cases} |x| & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fabs}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } x < 0 \\ 0 & \mbox{if } x = 0 \\ 1 & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Input variable.

Returns

Absolute value of variable.

Definition at line 50 of file fabs.hpp.

template<typename T >

bool stan::math::factor_cov_matrix	(	const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	Sigma,
		Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs,
		Eigen::Array< T, Eigen::Dynamic, 1 > &	sds
	)

This function is intended to make starting values, given a covariance matrix Sigma.

The transformations are hard coded as log for standard deviations and Fisher transformations (atanh()) of CPCs

Template Parameters

T	type of elements in the matrix and arrays

Parameters

[in]	Sigma	covariance matrix
[out]	CPCs	fill this unbounded (does not resize)
[out]	sds	fill this unbounded (does not resize)

Returns

false if any of the diagonals of Sigma are 0

Definition at line 25 of file factor_cov_matrix.hpp.

template<typename T >

void stan::math::factor_U	(	const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	U,
		Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs
	)

This function is intended to make starting values, given a unit upper-triangular matrix U such that U'DU is a correlation matrix.

Template Parameters

T	type of elements in the matrix

Parameters

U	Sigma matrix
CPCs	fill this unbounded

Definition at line 23 of file factor_U.hpp.

var stan::math::falling_factorial ( const var &  a, int  b  )

inline

Definition at line 25 of file falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::falling_factorial ( const fvar< T > &  x, int  n  )

inline

Return autodiff variable with the gradient and result of the falling factorial function applied to the inputs.

Will throw for NaN x and for negative n, as implemented in primitive function.

Template Parameters

T	Scalar type of autodiff variable.

Parameters

x	Argument.
n	Argument

Returns

tangent of falling factorial at arguments.

Definition at line 27 of file falling_factorial.hpp.

template<typename T >

return_type_t<T> stan::math::falling_factorial ( const T &  x, int  n  )

inline

Return the falling factorial function evaluated at the inputs.

Will throw for NaN x and for negative n

Template Parameters

T	Type of x argument.

Parameters

x	Argument.
n	Argument

Returns

Result of falling factorial function.

Exceptions

std::domain_error	if x is NaN
std::domain_error	if n is negative

\[ \mbox{falling\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ (x)_n & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{falling\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, (x)_n}{\partial x} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{falling\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, (x)_n}{\partial n} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ (x)_n=\frac{\Gamma(x+1)}{\Gamma(x-n+1)} \]

\[ \frac{\partial \, (x)_n}{\partial x} = (x)_n\Psi(x+1) \]

\[ \frac{\partial \, (x)_n}{\partial n} = -(x)_n\Psi(n+1) \]

Definition at line 63 of file falling_factorial.hpp.

template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

double stan::math::fdim ( T1  x, T2  y  )

inline

Return the positive difference of the specified values (C++11).

The function is defined by

fdim(x, y) = (x > y) ? (x - y) : 0.

Parameters

x	First value.
y	Second value.

Returns

max(x- y, 0)

Definition at line 21 of file fdim.hpp.

template<typename T >

fvar<T> stan::math::fdim ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return the positive difference of the specified values (C++11).

Template Parameters

T	Scalar type of autodiff variables.

Parameters

x	First argument.
y	Second argument.

Returns

Return the differences of the arguments if it is positive and 0 otherwise.

Definition at line 21 of file fdim.hpp.

template<typename T >

fvar<T> stan::math::fdim ( const fvar< T > &  x, double  y  )

inline

Return the positive difference of the specified values (C++11).

Template Parameters

T	Scalar type of autodiff variables.

Parameters

x	First argument.
y	Second argument.

Returns

Return the differences of the arguments if it is positive and 0 otherwise.

Definition at line 39 of file fdim.hpp.

template<typename T >

fvar<T> stan::math::fdim ( double  x, const fvar< T > &  y  )

inline

Return the positive difference of the specified values (C++11).

Template Parameters

T	Scalar type of autodiff variables.

Parameters

x	First argument.
y	Second argument.

Returns

Return the differences of the arguments if it is positive and 0 otherwise.

Definition at line 57 of file fdim.hpp.

var stan::math::fdim ( const var &  a, const var &  b  )

inline

Return the positive difference between the first variable's the value and the second's (C99, C++11).

The function values and deriatives are defined by

\[ \mbox{fdim}(x, y) = \begin{cases} x-y & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fdim}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fdim}(x, y)}{\partial y} = \begin{cases} -1 & \mbox{if } x > y \\[6pt] 0 & \mbox{otherwise} \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

The positive difference between the first and second variable.

Definition at line 91 of file fdim.hpp.

var stan::math::fdim ( double  a, const var &  b  )

inline

Return the positive difference between the first value and the value of the second variable (C99, C++11).

See fdim(var, var) for definitions of values and derivatives.

Parameters

a	First value.
b	Second variable.

Returns

The positive difference between the first and second arguments.

Definition at line 110 of file fdim.hpp.

var stan::math::fdim ( const var &  a, double  b  )

inline

Return the positive difference between the first variable's value and the second value (C99, C++11).

See fdim(var, var) for definitions of values and derivatives.

Parameters

a	First value.
b	Second variable.

Returns

The positive difference between the first and second arguments.

Definition at line 127 of file fdim.hpp.

template<typename T , typename S >

void stan::math::fill	(	T &	x,
		const S &	y
	)

Fill the specified container with the specified value.

This base case simply assigns the value to the container.

Template Parameters

T	Type of reference container.
S	Type of value.

Parameters

x	Container.
y	Value.

Definition at line 19 of file fill.hpp.

template<typename T , int R, int C, typename S >

void stan::math::fill	(	Eigen::Matrix< T, R, C > &	x,
		const S &	y
	)

Fill the specified container with the specified value.

The specified matrix is filled by element.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic
S	Type of value.

Parameters

x	Container.
y	Value.

Definition at line 23 of file fill.hpp.

template<typename T , typename S >

void stan::math::fill	(	std::vector< T > &	x,
		const S &	y
	)

Fill the specified container with the specified value.

Each container in the specified standard vector is filled recursively by calling fill.

Template Parameters

T	Type of container in vector.
S	Type of value.

Parameters

[in]	x	Container.
[in,out]	y	Value.

Definition at line 23 of file fill.hpp.

template<typename F >

void stan::math::finite_diff_grad_hessian	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::MatrixXd &	hess,
		std::vector< Eigen::MatrixXd > &	grad_hess_fx,
		double	epsilon = 1e-04
	)

Calculate the value and the gradient of the hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, Eigen::Dynamic, 1>&)

Reference:

De Levie: An improved numerical approximation for the first derivative, page 3

4 calls to the function, f.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	hess	Hessian matrix
[out]	grad_hess_fx	gradient of Hessian of function at argument
[in]	epsilon	perturbation size

Definition at line 41 of file finite_diff_grad_hessian.hpp.

template<typename F >

void stan::math::finite_diff_grad_hessian_auto	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::MatrixXd &	hess,
		std::vector< Eigen::MatrixXd > &	grad_hess_fx
	)

Calculate the value, Hessian, and the gradient of the Hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference.

The functor must implement

double operator()(const Eigen::VectorXd&) const;

Reference for finite difference to compute gradient:

De Levie: An improved numerical approximation for the first derivative, page 3

Step size for dimension i is set automatically using stan::math::finite_diff_stepsize(H(i, j)); the nested finite differences are over entries in the Hessian.

Evaluating this function involves 6 calls to the Hessian autodiff function for each entry in the Hessian.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	hess	Hessian matrix
[out]	grad_hess_fx	gradient of Hessian of function at argument

Definition at line 44 of file finite_diff_grad_hessian_auto.hpp.

template<typename F >

void stan::math::finite_diff_gradient	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::VectorXd &	grad_fx,
		double	epsilon = 1e-03
	)

Calculate the value and the gradient of the specified function at the specified argument using finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, -1, 1>&) const;

If epsilon is chosen to be near the square root of the machine precision and the input vector elements are all roughly unit scale, and if the function has reasonable limits on variation, error should be on the order of epsilon^6.

The reference for the algorithm is:

Robert de Levie. 2009. An improved numerical approximation for the first derivative. Journal of Chemical Sciences 121(5), page 3.

Evaluating this function involves 6 calls to f for each dimension of the input.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument
[in]	epsilon	perturbation size

Definition at line 43 of file finite_diff_gradient.hpp.

template<typename F >

void stan::math::finite_diff_gradient_auto	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::VectorXd &	grad_fx
	)

Calculate the value and the gradient of the specified function at the specified argument using finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, -, 1>&) const;

Error of derivative in dimension i should be on the should be on order of epsilon(i)^6, where epsilon(i) = sqrt(delta) * abs(x(i)) for input x at dimension i.

The reference for this algorithm is:

Robert de Levie. 2009. An improved numerical approximation for the first derivative. Journal of Chemical Sciences 121(5), page 3.

The reference for automatically setting the difference is this section of the Wikipedia,

Numerical differentiation: practical considerations using floating point arithmetic.

Evaluating this function involves 6 calls to the function being differentiated for each dimension in the input, plus one global evaluation. All evaluations will be for double-precision inputs.

Template Parameters

F	Type of function

Parameters

[in]	f	function
[in]	x	argument to function
[out]	fx	function applied to argument
[out]	grad_fx	gradient of function at argument

Definition at line 50 of file finite_diff_gradient_auto.hpp.

template<typename F >

void stan::math::finite_diff_hessian	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::VectorXd &	grad_fx,
		Eigen::MatrixXd &	hess_fx,
		double	epsilon = 1e-03
	)

Calculate the value and the Hessian of the specified function at the specified argument using second-order finite difference with the specified perturbation step size.

The functor must implement double operator()(const Eigen::VectorXd&) const;

For details of the algorithm, see
Eberly, D., 2008. Derivative approximation by finite differences. Magic Software, Inc., Page 6.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument
[out]	hess_fx	Hessian of function at argument
[in]	epsilon	perturbation step size

Definition at line 36 of file finite_diff_hessian.hpp.

template<typename F >

void stan::math::finite_diff_hessian_auto	(	const F &	f,
		const Eigen::VectorXd &	x,
		double &	fx,
		Eigen::VectorXd &	grad_fx,
		Eigen::MatrixXd &	hess_fx
	)

Calculate the value and the Hessian of the specified function at the specified argument using second-order finite difference, automatically setting the stepsize between the function evaluations along a dimension.

The functor must implement

double operator()(const Eigen::VectorXd&)

For details of the algorithm, see
Eberly, D., 2008. Derivative approximation by finite differences. Magic Software, Inc., Page 6.

Step size for dimension i is set automatically using stan::math::finite_diff_stepsize(x(i)).

For each non-diagonal entry in the Hessian, the function is evaluated 16 times; the diagonal entries require 4 function evaluations.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument
[out]	hess_fx	Hessian of function at argument

Definition at line 42 of file finite_diff_hessian_auto.hpp.

template<typename F >

double stan::math::finite_diff_hessian_helper	(	const F &	f,
		const Eigen::VectorXd &	x,
		int	i,
		double	epsilon = 1e-03
	)

Return the subcalculation required by finite_diff_hessian and finite_diff_hessian_auto.

The calculation is like a partial derivative calculation, but returns different values. The only utility of this function is as a subroutine for the two Hessians.

The functor must implement double operator()(const Eigen::VectorXd&) const;

This function evaluations the functor four times.

Template Parameters

F	type of function

Parameters

f	function to differentiate
x	argument to function
i	dimension of argument for derivative
epsilon	step size for finite differences

Returns

derivative of f(x) with respect to x(i)

Definition at line 30 of file finite_diff_hessian_helper.hpp.

double stan::math::finite_diff_stepsize ( double  u )

inline

Return the stepsize for finite difference evaluations at the specified scalar.

The forumula used is stepsize(u) = cbrt(epsilon) * max(1, abs(u)).

Parameters

u	initial value to increment

Returns

stepsize away from u for finite differences

Definition at line 20 of file finite_diff_stepsize.hpp.

template<typename T >

fvar<T> stan::math::floor ( const fvar< T > &  x )

inline

Definition at line 12 of file floor.hpp.

template<typename T >

apply_scalar_unary<floor_fun, T>::return_t stan::math::floor ( const T &  x )

inline

Vectorized version of floor().

Template Parameters

T	type of container

Parameters

x

container

Returns

Greatest integer <= each value in x.

Definition at line 33 of file floor.hpp.

var stan::math::floor ( const var &  a )

inline

Return the floor of the specified variable (cmath).

The derivative of the floor function is defined and zero everywhere but at integers, so we set these derivatives to zero for convenience,

\(\frac{d}{dx} {\lfloor x \rfloor} = 0\).

The floor function rounds down. For double values, this is the largest integral value that is not greater than the specified value. Although this function is not differentiable because it is discontinuous at integral values, its gradient is returned as zero everywhere.

\[ \mbox{floor}(x) = \begin{cases} \lfloor x \rfloor & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{floor}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Input variable.

Returns

Floor of the variable.

Definition at line 59 of file floor.hpp.

template<typename T1 , typename T2 , typename T3 , typename = require_all_arithmetic_t<T1, T2, T3>>

double stan::math::fma ( T1  x, T2  y, T3  z  )

inline

Return the product of the first two arguments plus the third argument.

Warning: This does not delegate to the high-precision platform-specific fma() implementation.

Parameters

x	First argument.
y	Second argument.
z	Third argument.

Returns

The product of the first two arguments plus the third argument.

Definition at line 25 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const fvar< T1 > &  x1, const fvar< T2 > &  x2, const fvar< T3 > &  x3  )

inline

The fused multiply-add operation (C99).

This double-based operation delegates to fma.

The function is defined by

fma(a, b, c) = (a * b) + c.

\[ \mbox{fma}(x, y, z) = \begin{cases} x\cdot y+z & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial x} = \begin{cases} y & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial y} = \begin{cases} x & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial z} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

x1	First value.
x2	Second value.
x3	Third value.

Returns

Product of the first two values plus the third.

Definition at line 59 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const T1 &  x1, const fvar< T2 > &  x2, const fvar< T3 > &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 70 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const fvar< T1 > &  x1, const T2 &  x2, const fvar< T3 > &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 80 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const fvar< T1 > &  x1, const fvar< T2 > &  x2, const T3 &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 90 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const T1 &  x1, const T2 &  x2, const fvar< T3 > &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 100 of file fma.hpp.

var stan::math::fma ( const var &  a, const var &  b, const var &  c  )

inline

The fused multiply-add function for three variables (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivatives are

\(\frac{\partial}{\partial x} (x * y) + z = y\), and

\(\frac{\partial}{\partial y} (x * y) + z = x\), and

\(\frac{\partial}{\partial z} (x * y) + z = 1\).

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 106 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const fvar< T1 > &  x1, const T2 &  x2, const T3 &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 110 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const T1 &  x1, const fvar< T2 > &  x2, const T3 &  x3  )

inline

See all-var input signature for details on the function and derivatives.

Definition at line 120 of file fma.hpp.

template<typename Tc , typename = require_arithmetic_t<Tc>>

var stan::math::fma ( const var &  a, const var &  b, Tc &&  c  )

inline

The fused multiply-add function for two variables and a value (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivatives are

\(\frac{\partial}{\partial x} (x * y) + c = y\), and

\(\frac{\partial}{\partial y} (x * y) + c = x\).

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 127 of file fma.hpp.

template<typename Ta , typename Tb , typename Tc , typename = require_arithmetic_t<Tb>, typename = require_all_var_t<Ta, Tc>>

var stan::math::fma ( Ta &&  a, Tb &&  b, Tc &&  c  )

inline

The fused multiply-add function for a variable, value, and variable (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivatives are

\(\frac{\partial}{\partial x} (x * c) + z = c\), and

\(\frac{\partial}{\partial z} (x * c) + z = 1\).

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 150 of file fma.hpp.

template<typename Tb , typename Tc , typename = require_all_arithmetic_t<Tb, Tc>>

var stan::math::fma ( const var &  a, Tb &&  b, Tc &&  c  )

inline

The fused multiply-add function for a variable and two values (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The derivative is

\(\frac{d}{d x} (x * c) + d = c\).

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 173 of file fma.hpp.

template<typename Ta , typename Tc , typename = require_all_arithmetic_t<Ta, Tc>>

var stan::math::fma ( Ta &&  a, const var &  b, Tc &&  c  )

inline

The fused multiply-add function for a value, variable, and value (C99).

This function returns the product of the first two arguments plus the third argument.

The derivative is

\(\frac{d}{d y} (c * y) + d = c\), and

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 192 of file fma.hpp.

template<typename Ta , typename Tb , typename = require_all_arithmetic_t<Ta, Tb>>

var stan::math::fma ( Ta &&  a, Tb &&  b, const var &  c  )

inline

The fused multiply-add function for two values and a variable, and value (C99).

This function returns the product of the first two arguments plus the third argument.

The derivative is

\(\frac{\partial}{\partial z} (c * d) + z = 1\).

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 211 of file fma.hpp.

template<typename Ta , typename = require_arithmetic_t<Ta>>

var stan::math::fma ( Ta &&  a, const var &  b, const var &  c  )

inline

The fused multiply-add function for a value and two variables (C99).

This function returns the product of the first two arguments plus the third argument.

The partial derivaties are

\(\frac{\partial}{\partial y} (c * y) + z = c\), and

\(\frac{\partial}{\partial z} (c * y) + z = 1\).

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns

Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 232 of file fma.hpp.

template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

double stan::math::fmax ( T1  x, T2  y  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Parameters

x	First argument.
y	Second argument.

Returns

maximum of x or y and if one is NaN return the other

Definition at line 19 of file fmax.hpp.

template<typename T >

fvar<T> stan::math::fmax ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Parameters

x1	First argument.
x2	Second argument.

Returns

maximum of arguments, and if one is NaN return the other

Definition at line 22 of file fmax.hpp.

template<typename T >

fvar<T> stan::math::fmax ( double  x1, const fvar< T > &  x2  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Parameters

x1	First argument.
x2	Second argument.

Returns

maximum of arguments, and if one is NaN return the other

Definition at line 49 of file fmax.hpp.

var stan::math::fmax ( const var &  a, const var &  b  )

inline

Returns the maximum of the two variable arguments (C99).

No new variable implementations are created, with this function defined as if by

fmax(a, b) = a if a's value is greater than b's, and .

fmax(a, b) = b if b's value is greater than or equal to a's.

\[ \mbox{fmax}(x, y) = \begin{cases} x & \mbox{if } x \geq y \\ y & \mbox{if } x < y \\[6pt] x & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ y & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmax}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x \geq y \\ 0 & \mbox{if } x < y \\[6pt] 1 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 0 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmax}(x, y)}{\partial y} = \begin{cases} 0 & \mbox{if } x \geq y \\ 1 & \mbox{if } x < y \\[6pt] 0 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 1 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

If the first variable's value is larger than the second's, the first variable, otherwise the second variable.

Definition at line 61 of file fmax.hpp.

template<typename T >

fvar<T> stan::math::fmax ( const fvar< T > &  x1, double  x2  )

inline

Return the greater of the two specified arguments.

If one is not-a-number, return the other.

Parameters

x1	First argument.
x2	Second argument.

Returns

maximum of arguments, and if one is NaN return the other

Definition at line 76 of file fmax.hpp.

var stan::math::fmax ( const var &  a, double  b  )

inline

Returns the maximum of the variable and scalar, promoting the scalar to a variable if it is larger (C99).

For fmax(a, b), if a's value is greater than b, then a is returned, otherwise a fesh variable implementation wrapping the value b is returned.

Parameters

a	First variable.
b	Second value

Returns

If the first variable's value is larger than or equal to the second value, the first variable, otherwise the second value promoted to a fresh variable.

Definition at line 89 of file fmax.hpp.

var stan::math::fmax ( double  a, const var &  b  )

inline

Returns the maximum of a scalar and variable, promoting the scalar to a variable if it is larger (C99).

For fmax(a, b), if a is greater than b's value, then a fresh variable implementation wrapping a is returned, otherwise b is returned.

Parameters

a	First value.
b	Second variable.

Returns

If the first value is larger than the second variable's value, return the first value promoted to a variable, otherwise return the second variable.

Definition at line 116 of file fmax.hpp.

template<typename T >

fvar<T> stan::math::fmin ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file fmin.hpp.

template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

double stan::math::fmin ( T1  x, T2  y  )

inline

Return the lesser of the two specified arguments.

If one is greater than the other, return not-a-number.

Parameters

x	First argument.
y	Second argument.

Returns

Minimum of x or y and if one is NaN return the other

Definition at line 19 of file fmin.hpp.

template<typename T >

fvar<T> stan::math::fmin ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 33 of file fmin.hpp.

template<typename T >

fvar<T> stan::math::fmin ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 52 of file fmin.hpp.

var stan::math::fmin ( const var &  a, const var &  b  )

inline

Returns the minimum of the two variable arguments (C99).

For fmin(a, b), if a's value is less than b's, then a is returned, otherwise b is returned.

\[ \mbox{fmin}(x, y) = \begin{cases} x & \mbox{if } x \leq y \\ y & \mbox{if } x > y \\[6pt] x & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ y & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmin}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x \leq y \\ 0 & \mbox{if } x > y \\[6pt] 1 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 0 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmin}(x, y)}{\partial y} = \begin{cases} 0 & \mbox{if } x \leq y \\ 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 1 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

If the first variable's value is smaller than the second's, the first variable, otherwise the second variable.

Definition at line 57 of file fmin.hpp.

var stan::math::fmin ( const var &  a, double  b  )

inline

Returns the minimum of the variable and scalar, promoting the scalar to a variable if it is larger (C99).

For fmin(a, b), if a's value is less than or equal to b, then a is returned, otherwise a fresh variable wrapping b is returned.

Parameters

a	First variable.
b	Second value

Returns

If the first variable's value is less than or equal to the second value, the first variable, otherwise the second value promoted to a fresh variable.

Definition at line 85 of file fmin.hpp.

var stan::math::fmin ( double  a, const var &  b  )

inline

Returns the minimum of a scalar and variable, promoting the scalar to a variable if it is larger (C99).

For fmin(a, b), if a is less than b's value, then a fresh variable implementation wrapping a is returned, otherwise b is returned.

Parameters

a	First value.
b	Second variable.

Returns

If the first value is smaller than the second variable's value, return the first value promoted to a variable, otherwise return the second variable.

Definition at line 112 of file fmin.hpp.

template<typename T >

fvar<T> stan::math::fmod ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 15 of file fmod.hpp.

template<typename T >

fvar<T> stan::math::fmod ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 23 of file fmod.hpp.

template<typename T >

fvar<T> stan::math::fmod ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 33 of file fmod.hpp.

var stan::math::fmod ( const var &  a, const var &  b  )

inline

Return the floating point remainder after dividing the first variable by the second (cmath).

The partial derivatives with respect to the variables are defined everywhere but where \(x = y\), but we set these to match other values, with

\(\frac{\partial}{\partial x} \mbox{fmod}(x, y) = 1\), and

\(\frac{\partial}{\partial y} \mbox{fmod}(x, y) = -\lfloor \frac{x}{y} \rfloor\).

\[ \mbox{fmod}(x, y) = \begin{cases} x - \lfloor \frac{x}{y}\rfloor y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmod}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmod}(x, y)}{\partial y} = \begin{cases} -\lfloor \frac{x}{y}\rfloor & \mbox{if } -\infty\leq x, y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

Floating pointer remainder of dividing the first variable by the second.

Definition at line 100 of file fmod.hpp.

var stan::math::fmod ( const var &  a, double  b  )

inline

Return the floating point remainder after dividing the the first variable by the second scalar (cmath).

The derivative with respect to the variable is

\(\frac{d}{d x} \mbox{fmod}(x, c) = \frac{1}{c}\).

Parameters

a	First variable.
b	Second scalar.

Returns

Floating pointer remainder of dividing the first variable by the second scalar.

Definition at line 117 of file fmod.hpp.

var stan::math::fmod ( double  a, const var &  b  )

inline

Return the floating point remainder after dividing the first scalar by the second variable (cmath).

The derivative with respect to the variable is

\(\frac{d}{d y} \mbox{fmod}(c, y) = -\lfloor \frac{c}{y} \rfloor\).

Parameters

a	First scalar.
b	Second variable.

Returns

Floating pointer remainder of dividing first scalar by the second variable.

Definition at line 134 of file fmod.hpp.

template<typename T_desired , typename T_actual , typename = std::enable_if_t< !std::is_convertible<T_actual, T_desired>::value || static_cast<int>(T_desired::RowsAtCompileTime) != static_cast<int>(T_actual::RowsAtCompileTime) || static_cast<int>(T_desired::ColsAtCompileTime) != static_cast<int>(T_actual::ColsAtCompileTime)>, typename = void>

T_desired stan::math::forward_as ( const T_actual &  a )

inline

Assume which type we get.

If actual type is not convertible to assumed type or in case of eigen types compile time rows and columns are not the same this has return type of T_desired, but it only throws. This version should only be used where it is optimized away so the throw should never happen.

This is intended to be used in compile time branches that would otherwise trigger compile error even though they are never executed.

Template Parameters

T_desired	type of output we need to avoid compile time errors
T_actual	actual type of the argument

Parameters

a	input value

Returns

nothing, this always throws

Exceptions

always

throws std::runtime_error

Definition at line 105 of file forward_as.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_cdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	sigma
	)

Definition at line 19 of file frechet_cdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_lccdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	sigma
	)

Definition at line 19 of file frechet_lccdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_lcdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	sigma
	)

Definition at line 19 of file frechet_lcdf.hpp.

template<bool propto, typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_lpdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	sigma
	)

Definition at line 21 of file frechet_lpdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::frechet_lpdf ( const T_y &  y, const T_shape &  alpha, const T_scale &  sigma  )

inline

Definition at line 126 of file frechet_lpdf.hpp.

template<int R = Eigen::Dynamic, int C = Eigen::Dynamic, typename T , require_var_t< T > ...>

Eigen::Matrix<double, R, C> stan::math::from_matrix_cl ( const matrix_cl< T > &  src )

inline

Copies the adjoint of the source matrix of vars that is stored on the OpenCL device to the destination Eigen matrix.

Parameters

src	source matrix of vars on the OpenCL device

Returns

Eigen matrix with a copy of the adjoints in the source matrix

Definition at line 54 of file copy.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >

return_type_t<T_y, T_shape, T_inv_scale> stan::math::gamma_lccdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 19 of file gamma_lccdf.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >

return_type_t<T_y, T_shape, T_inv_scale> stan::math::gamma_lcdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 19 of file gamma_lcdf.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >

return_type_t<T_y, T_shape, T_inv_scale> stan::math::gamma_lpdf ( const T_y &  y, const T_shape &  alpha, const T_inv_scale &  beta  )

inline

Definition at line 145 of file gamma_lpdf.hpp.

template<typename T >

fvar<T> stan::math::gamma_p ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 16 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::math::gamma_p ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 35 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::math::gamma_p ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 50 of file gamma_p.hpp.

double stan::math::gamma_p ( double  z, double  a  )

inline

Return the value of the normalized, lower-incomplete gamma function applied to the specified argument.

This function is defined, including error conditions, as follows

\[ \mbox{gamma\_p}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ P(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_p}(a, z)}{\partial a} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, P(a, z)}{\partial a} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_p}(a, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, P(a, z)}{\partial z} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ P(a, z)=\frac{1}{\Gamma(a)}\int_0^zt^{a-1}e^{-t}dt \]

\[ \frac{\partial \, P(a, z)}{\partial a} = -\frac{\Psi(a)}{\Gamma^2(a)}\int_0^zt^{a-1}e^{-t}dt + \frac{1}{\Gamma(a)}\int_0^z (a-1)t^{a-2}e^{-t}dt \]

\[ \frac{\partial \, P(a, z)}{\partial z} = \frac{z^{a-1}e^{-z}}{\Gamma(a)} \]

Parameters

z	first argument
a	second argument

Returns

value of the normalized, lower-incomplete gamma function applied to z and a

Exceptions

std::domain_error

if either argument is not positive or if z is at a pole of the function

Definition at line 66 of file gamma_p.hpp.

var stan::math::gamma_p ( const var &  a, const var &  b  )

inline

Definition at line 102 of file gamma_p.hpp.

var stan::math::gamma_p ( const var &  a, double  b  )

inline

Definition at line 106 of file gamma_p.hpp.

var stan::math::gamma_p ( double  a, const var &  b  )

inline

Definition at line 110 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::math::gamma_q ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file gamma_q.hpp.

template<typename T >

fvar<T> stan::math::gamma_q ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 46 of file gamma_q.hpp.

var stan::math::gamma_q ( const var &  a, const var &  b  )

inline

Definition at line 50 of file gamma_q.hpp.

double stan::math::gamma_q ( double  x, double  a  )

inline

\[ \mbox{gamma\_q}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ Q(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_q}(a, z)}{\partial a} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, Q(a, z)}{\partial a} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_q}(a, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, Q(a, z)}{\partial z} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ Q(a, z)=\frac{1}{\Gamma(a)}\int_z^\infty t^{a-1}e^{-t}dt \]

\[ \frac{\partial \, Q(a, z)}{\partial a} = -\frac{\Psi(a)}{\Gamma^2(a)}\int_z^\infty t^{a-1}e^{-t}dt + \frac{1}{\Gamma(a)}\int_z^\infty (a-1)t^{a-2}e^{-t}dt \]

\[ \frac{\partial \, Q(a, z)}{\partial z} = -\frac{z^{a-1}e^{-z}}{\Gamma(a)} \]

Exceptions

domain_error

if x is at pole

Definition at line 54 of file gamma_q.hpp.

var stan::math::gamma_q ( const var &  a, double  b  )

inline

Definition at line 54 of file gamma_q.hpp.

var stan::math::gamma_q ( double  a, const var &  b  )

inline

Definition at line 58 of file gamma_q.hpp.

template<typename T >

fvar<T> stan::math::gamma_q ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 77 of file gamma_q.hpp.

template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >

return_type_t<T_y, return_type_t<T_F, T_G, T_V, T_W, T_m0, T_C0> > stan::math::gaussian_dlm_obs_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &  F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &  G, const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &  V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &  W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &  m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &  C0  )

inline

Definition at line 165 of file gaussian_dlm_obs_lpdf.hpp.

template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >

return_type_t<T_y, return_type_t<T_F, T_G, T_V, T_W, T_m0, T_C0> > stan::math::gaussian_dlm_obs_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &  F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &  G, const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &  V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &  W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &  m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &  C0  )

inline

Definition at line 323 of file gaussian_dlm_obs_lpdf.hpp.

template<class RNG >

Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::gaussian_dlm_obs_rng ( const Eigen::MatrixXd &  F, const Eigen::MatrixXd &  G, const Eigen::VectorXd &  V, const Eigen::MatrixXd &  W, const Eigen::VectorXd &  m0, const Eigen::MatrixXd &  C0, const int  T, RNG &  rng  )

inline

Definition at line 177 of file gaussian_dlm_obs_rng.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< T > &  x, size_t  i, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i	Index into vector plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at i - 1

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 27 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< T > > &  x, size_t  i1, size_t  i2, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 50 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< T > > > &  x, size_t  i1, size_t  i2, size_t  i3, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 74 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< T > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 100 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 128 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 158 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, size_t  i7, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
i7	Seventh index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 189 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, size_t  i7, size_t  i8, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
i7	Seventh index plus 1.
i8	Eigth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 221 of file get_base1.hpp.

template<typename T >

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::get_base1 ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x, size_t  m, const char *  error_msg, size_t  idx  )

inline

Return a copy of the row of the specified vector at the specified base-one row index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Warning: Because a copy is involved, it is inefficient to access element of matrices by first using this method to get a row then using a second call to get the value at a specified column.

Template Parameters

T	type of value

Parameters

x	Matrix from which to get a row
m	Index into matrix plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Row of matrix at i - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 251 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x, size_t  m, size_t  n, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified matrix at the specified base-one row and column indexes.

If either index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Matrix from which to get a row
m	Row index plus 1.
n	Column index plus 1.
error_msg	Error message if either index is out of range.
idx	Nested index level to report in error message if either index is out of range.

Returns

Value of matrix at row m - 1 and column n - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 276 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x, size_t  m, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified column vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Column vector from which to get a value.
m	Row index plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of column vector at row m - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 300 of file get_base1.hpp.

template<typename T >

const T& stan::math::get_base1 ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  x, size_t  n, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified row vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Row vector from which to get a value.
n	Column index plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of row vector at column n - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 322 of file get_base1.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< T > &  x, size_t  i, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i	Index into vector plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at i - 1

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 27 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< T > > &  x, size_t  i1, size_t  i2, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 50 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< T > > > &  x, size_t  i1, size_t  i2, size_t  i3, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 74 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< T > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 100 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 127 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 156 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, size_t  i7, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
i7	Seventh index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 187 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &  x, size_t  i1, size_t  i2, size_t  i3, size_t  i4, size_t  i5, size_t  i6, size_t  i7, size_t  i8, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Vector from which to get a value.
i1	First index plus 1.
i2	Second index plus 1.
i3	Third index plus 1.
i4	Fourth index plus 1.
i5	Fifth index plus 1.
i6	Sixth index plus 1.
i7	Seventh index plus 1.
i8	Eigth index plus 1.
error_msg	Error message if an index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of vector at indexes.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 219 of file get_base1_lhs.hpp.

template<typename T >

Eigen::Block<Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> > stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x, size_t  m, const char *  error_msg, size_t  idx  )

inline

Return a copy of the row of the specified vector at the specified base-one row index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Warning: Because a copy is involved, it is inefficient to access element of matrices by first using this method to get a row then using a second call to get the value at a specified column.

Template Parameters

T	type of value

Parameters

x	Matrix from which to get a row
m	Index into matrix plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Row of matrix at i - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 251 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x, size_t  m, size_t  n, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified matrix at the specified base-one row and column indexes.

If either index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Matrix from which to get a row
m	Row index plus 1.
n	Column index plus 1.
error_msg	Error message if either index is out of range.
idx	Nested index level to report in error message if either index is out of range.

Returns

Value of matrix at row m - 1 and column n - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 275 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x, size_t  m, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified column vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Column vector from which to get a value.
m	Row index plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of column vector at row m - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 298 of file get_base1_lhs.hpp.

template<typename T >

T& stan::math::get_base1_lhs ( Eigen::Matrix< T, 1, Eigen::Dynamic > &  x, size_t  n, const char *  error_msg, size_t  idx  )

inline

Return a reference to the value of the specified row vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Template Parameters

T	type of value

Parameters

x	Row vector from which to get a value.
n	Column index plus 1.
error_msg	Error message if the index is out of range.
idx	Nested index level to report in error message if the index is out of range.

Returns

Value of row vector at column n - 1.

Exceptions

std::out_of_range

if idx is out of range.

Definition at line 320 of file get_base1_lhs.hpp.

template<typename T_lp , typename T_lp_accum >

return_type_t<T_lp, T_lp_accum> stan::math::get_lp ( const T_lp &  lp, const accumulator< T_lp_accum > &  lp_accum  )

inline

Definition at line 10 of file get_lp.hpp.

template<typename T_x , typename T_sigma >

Eigen::Matrix<return_type_t<T_x, T_sigma>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_dot_prod_cov	(	const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &	x,
		const T_sigma &	sigma
	)

Returns a dot product covariance matrix.

A member of Stan's Gaussian Process Library.

\(k(x,x') = \sigma^2 + x \cdot x'\)

A dot product covariance matrix is the same covariance matrix as in bayesian regression with \(N(0,1)\) priors on regression coefficients and a \(N(0,\sigma^2)\) prior on the constant function. See Rasmussen and Williams et al 2006, Chapter 4.

Template Parameters

T_x	type of std::vector of elements
T_sigma	type of sigma

Parameters

x	std::vector of elements that can be used in dot product. This function assumes each element of x is the same size.
sigma	constant function that can be used in stan::math::square

Returns

dot product covariance matrix that is positive semi-definite

Exceptions

std::domain_error

if sigma < 0, nan, inf or x is nan or infinite

Definition at line 38 of file gp_dot_prod_cov.hpp.

template<typename T_x , typename T_sigma >

Eigen::Matrix<return_type_t<T_x, T_sigma>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_dot_prod_cov	(	const std::vector< T_x > &	x,
		const T_sigma &	sigma
	)

Returns a dot product covariance matrix.

A member of Stan's Gaussian Process Library.

\(k(x,x') = \sigma^2 + x \cdot x'\)

A dot product covariance matrix is the same covariance matrix as in bayesian regression with \(N(0,1)\) priors on regression coefficients and a \(N(0,\sigma^2)\) prior on the constant function. See Rasmussen and Williams et al 2006, Chapter 4.

Template Parameters

T_x	type of std::vector of double
T_sigma	type of sigma

Parameters

x	std::vector of elements that can be used in transpose and multiply This function assumes each element of x is the same size.
sigma	constant function that can be used in stan::math::square

Returns

dot product covariance matrix that is positive semi-definite

Exceptions

std::domain_error

if sigma < 0, nan, inf or x is nan or infinite

Definition at line 93 of file gp_dot_prod_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_dot_prod_cov	(	const std::vector< Eigen::Matrix< T_x1, Eigen::Dynamic, 1 >> &	x1,
		const std::vector< Eigen::Matrix< T_x2, Eigen::Dynamic, 1 >> &	x2,
		const T_sigma &	sigma
	)

Returns a dot product covariance matrix of differing x's.

A member of Stan's Gaussian Process Library.

\(k(x,x') = \sigma^2 + x \cdot x'\)

A dot product covariance matrix is the same covariance matrix as in bayesian regression with \(N(0,1)\) priors on regression coefficients and a \(N(0,\sigma^2)\) prior on the constant function. See Rasmussen and Williams et al 2006, Chapter 4.

Template Parameters

T_x1	type of first std::vector of elements
T_x2	type of second std::vector of elements
T_sigma	type of sigma

Parameters

x1	std::vector of elements that can be used in dot_product
x2	std::vector of elements that can be used in dot_product
sigma	constant function that can be used in stan::math::square

Returns

dot product covariance matrix that is not always symmetric

Exceptions

std::domain_error

if sigma < 0, nan or inf or if x1 or x2 are nan or inf

Definition at line 146 of file gp_dot_prod_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_dot_prod_cov	(	const std::vector< T_x1 > &	x1,
		const std::vector< T_x2 > &	x2,
		const T_sigma &	sigma
	)

Returns a dot product covariance matrix of differing x's.

A member of Stan's Gaussian Process Library.

\(k(x,x') = \sigma^2 + x \cdot x'\)

A dot product covariance matrix is the same covariance matrix as in bayesian regression with \(N(0,1)\) priors on regression coefficients and a \(N(0,\sigma^2)\) prior on the constant function. See Rasmussen and Williams et al 2006, Chapter 4.

Template Parameters

T_x1	type of first std::vector of double
T_x2	type of second std::vector of double
T_sigma	type of sigma

Parameters

x1	std::vector of elements that can be used in dot_product
x2	std::vector of elements that can be used in dot_product
sigma	is the constant function that can be used in stan::math::square

Returns

dot product covariance matrix that is not always symmetric

Exceptions

std::domain_error

if sigma < 0, nan or inf or if x1 or x2 are nan or inf

Definition at line 206 of file gp_dot_prod_cov.hpp.

template<typename T_x , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exp_quad_cov ( const std::vector< T_x > &  x, const T_sigma &  sigma, const T_l &  length_scale  )

inline

Returns a squared exponential kernel.

Template Parameters

T_x	type for each scalar
T_sigma	type of parameter sigma
T_l	type of parameter length scale

Parameters

x	std::vector of scalars that can be used in square distance. This function assumes each element of x is the same size.
sigma	marginal standard deviation or magnitude
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 175 of file gp_exp_quad_cov.hpp.

template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>

Eigen::Matrix<var, -1, -1> stan::math::gp_exp_quad_cov ( const std::vector< T_x > &  x, const var &  sigma, const var &  length_scale  )

inline

Returns a squared exponential kernel.

Parameters

x	std::vector input that can be used in square distance Assumes each element of x is the same size
sigma	standard deviation
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 212 of file gp_exp_quad_cov.hpp.

template<typename T_x , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exp_quad_cov ( const std::vector< Eigen::Matrix< T_x,-1, 1 >> &  x, const T_sigma &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a squared exponential kernel.

Template Parameters

T_x	type for each scalar
T_sigma	type of parameter sigma
T_l	type of each length scale parameter

Parameters

x	std::vector of Eigen vectors of scalars.
sigma	marginal standard deviation or magnitude
length_scale	std::vector length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 215 of file gp_exp_quad_cov.hpp.

template<typename T_x , typename = require_arithmetic_t<typename scalar_type<T_x>::type>>

Eigen::Matrix<var, -1, -1> stan::math::gp_exp_quad_cov ( const std::vector< T_x > &  x, double  sigma, const var &  length_scale  )

inline

Returns a squared exponential kernel.

Parameters

x	std::vector input that can be used in square distance Assumes each element of x is the same size
sigma	standard deviation
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 256 of file gp_exp_quad_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exp_quad_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_sigma &  sigma, const T_l &  length_scale  )

inline

Returns a squared exponential kernel.

This function is for the cross covariance matrix needed to compute posterior predictive density.

Template Parameters

T_x1	type of first std::vector of scalars
T_x2	type of second std::vector of scalars This function assumes each element of x1 and x2 are the same size.
T_sigma	type of sigma
T_l	type of of length scale

Parameters

x1	std::vector of elements that can be used in square distance
x2	std::vector of elements that can be used in square distance
sigma	standard deviation
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 258 of file gp_exp_quad_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exp_quad_cov ( const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &  x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &  x2, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a squared exponential kernel.

This function is for the cross covariance matrix needed to compute the posterior predictive density.

Template Parameters

T_x1	type of first std::vector of elements
T_x2	type of second std::vector of elements
T_s	type of sigma
T_l	type of length scale

Parameters

x1	std::vector of Eigen vectors of scalars.
x2	std::vector of Eigen vectors of scalars.
sigma	standard deviation
length_scale	std::vector of length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 307 of file gp_exp_quad_cov.hpp.

template<>

Eigen::MatrixXd stan::math::gp_exp_quad_cov ( const std::vector< double > &  x, const double &  sigma, const double &  length_scale  )

inline

Returns a squared exponential kernel.

Parameters

x	std::vector of scalars that can be used in square distance.
sigma	marginal standard deviation or magnitude
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 353 of file gp_exp_quad_cov.hpp.

template<>

Eigen::MatrixXd stan::math::gp_exp_quad_cov ( const std::vector< Eigen::VectorXd > &  x, const double &  sigma, const double &  length_scale  )

inline

Returns a squared exponential kernel.

Parameters

x	std::vector of scalars that can be used in square distance. This function assumes each element of x is the same size.
sigma	marginal standard deviation or magnitude
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 396 of file gp_exp_quad_cov.hpp.

template<>

Eigen::MatrixXd stan::math::gp_exp_quad_cov ( const std::vector< Eigen::VectorXd > &  x, const double &  sigma, const std::vector< double > &  length_scale  )

inline

Returns a squared exponential kernel.

Parameters

x	std::vector of Eigen vectors of scalars.
sigma	marginal standard deviation or magnitude
length_scale	std::vector length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 440 of file gp_exp_quad_cov.hpp.

template<>

Eigen::MatrixXd stan::math::gp_exp_quad_cov ( const std::vector< double > &  x1, const std::vector< double > &  x2, const double &  sigma, const double &  length_scale  )

inline

Returns a squared exponential kernel.

This function is for the cross covariance matrix needed to compute posterior predictive density.

Parameters

x1	std::vector of elements that can be used in square distance
x2	std::vector of elements that can be used in square distance
sigma	standard deviation
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 487 of file gp_exp_quad_cov.hpp.

template<>

Eigen::MatrixXd stan::math::gp_exp_quad_cov ( const std::vector< Eigen::VectorXd > &  x1, const std::vector< Eigen::VectorXd > &  x2, const double &  sigma, const double &  length_scale  )

inline

Returns a squared exponential kernel.

This function is for the cross covariance matrix needed to compute posterior predictive density.

Parameters

x1	std::vector of elements that can be used in square distance
x2	std::vector of elements that can be used in square distance This function assumes each element of x1 and x2 are the same size.
sigma	standard deviation
length_scale	length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 538 of file gp_exp_quad_cov.hpp.

template<>

Eigen::MatrixXd stan::math::gp_exp_quad_cov ( const std::vector< Eigen::VectorXd > &  x1, const std::vector< Eigen::VectorXd > &  x2, const double &  sigma, const std::vector< double > &  length_scale  )

inline

Returns a squared exponential kernel.

This function is for the cross covariance matrix needed to compute the posterior predictive density.

Parameters

x1	std::vector of Eigen vectors of scalars.
x2	std::vector of Eigen vectors of scalars.
sigma	standard deviation
length_scale	std::vector of length scale

Returns

squared distance

Exceptions

std::domain_error

if sigma <= 0, l <= 0, or x is nan or infinite

Definition at line 594 of file gp_exp_quad_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exponential_cov ( const std::vector< T_x > &  x, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern exponential covariance Matrix.

\[ k(x, x') = \sigma^2 exp(-\frac{d(x, x')^2}{2l^2}) \]

where d(x, x') is the Euclidean distance.

Template Parameters

T_x	type for each scalar
T_s	type of parameter sigma
T_l	type of parameter length scale

Parameters

x	std::vector of scalars that can be used in stan::math::distance
sigma	standard deviation or magnitude
length_scale	length scale

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 37 of file gp_exponential_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exponential_cov ( const std::vector< Eigen::Matrix< T_x,-1, 1 >> &  x, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern exponential covariance matrix.

\[ k(x, x') = \sigma^2 exp(-\sum_{k=1}^K\frac{d(x, x')}{l_k}) \]

where d(x, x') is the Euclidean distance.

Template Parameters

T_x	type for each scalar
T_s	type for each parameter sigma
T_l	type for each length scale parameter

Parameters

x	std::vector of Eigen::vectors of scalars
sigma	standard deviation that can be used in stan::math::square
length_scale	std::vector of length scales

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 95 of file gp_exponential_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exponential_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern exponential cross covariance matrix.

\[ k(x, x') = \sigma^2 exp(-\frac{d(x, x')}{l}) \]

where d(x, x') is the Euclidean distance This function is for the cross covariance matrix needed to compute the posterior predictive distribution

Template Parameters

T_x1	first type of scalars contained in vector x1
T_x2	second type of scalars contained in vector x2
T_s	type of parameter sigma, marginal standard deviation
T_l	type of parameter length scale

Parameters

x1	std::vector of scalars that can be used in squared_distance
x2	std::vector of scalars that can be used in squared_distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x1, x2 are nan or inf

Definition at line 157 of file gp_exponential_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_exponential_cov ( const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &  x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &  x2, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern exponential cross covariance matrix.

\[ k(x, x') = \sigma^2 exp(-\sum_{k=1}^K\frac{d(x, x')}{l_k}) \]

where \(d(x, x')\) is the Euclidean distance

This function is for the cross covariance matrix neededed to compute the posterior predictive density.

Template Parameters

T_x1	first type of std::vector of scalars
T_x2	second type of std::vector of scalars
T_s	type of parameter sigma, marginal standard deviation
T_l	type of parameter length scale

Parameters

x1	std::vector of Eigen vectors of scalars
x2	std::vector of Eigen vectors of scalars
length_scale	parameter length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x1, x2 are nan or inf

Definition at line 226 of file gp_exponential_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern32_cov ( const std::vector< T_x > &  x, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern 3/2 covariance matrix.

\[ k(x, x') = \sigma^2(1 + \frac{\sqrt{3}d(x, x')}{l})exp(-\frac{\sqrt{3}d(x, x')}{l}) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x	type for each scalar
T_s	type of parameter of sigma
T_l	type of parameter length scale

Parameters

x	std::vector of scalars that can be used in squared distance
length_scale	length scale
sigma	marginal standard deviation or magnitude

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 40 of file gp_matern32_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern32_cov ( const std::vector< Eigen::Matrix< T_x,-1, 1 >> &  x, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern 3/2 covariance matrix.

\[ k(x, x') = \sigma^2(1 + \sqrt{3} \sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}) exp(-\sqrt{3}\sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}) \]

where \(d(x, x')\) is the Euclidean distance.

Template Parameters

T_x	type for each scalar
T_s	type of element of parameter sigma
T_l	type of each length scale parameter

Parameters

x	std::vector of Eigen vectors of scalars.
length_scale	std::vector of length scales
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 105 of file gp_matern32_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern32_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern 3/2 cross covariance matrix.

\[ k(x, x') = \sigma^2(1 + \frac{\sqrt{3}d(x, x')}{l})exp(-\sqrt{3}\frac{d(x, x')}{l}) \]

where \(d(x, x')\) is the Euclidean distance.

This function is for the cross covariance matrix neededed to compute the posterior predictive density.

Template Parameters

T_x1	first type of scalars contained in vector x1
T_x2	second type of scalars contained in vector x2
T_s	type of parameter sigma, marginal standard deviation
T_l	type of parameter length scale

Parameters

x1	std::vector of scalars that can be used in squared_distance
x2	std::vector of scalars that can be used in squared_distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x1, x2 are nan or inf

Definition at line 176 of file gp_matern32_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern32_cov ( const std::vector< Eigen::Matrix< T_x1,-1, 1 >> &  x1, const std::vector< Eigen::Matrix< T_x2,-1, 1 >> &  x2, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern 3/2 cross covariance matrix.

\[ k(x, x') = \sigma^2(1 + \sqrt{3} \sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}) exp(-\sqrt{3}\sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}) \]

where \(d(x, x')\) is the Euclidean distance

This function is for the cross covariance matrix neededed to compute the posterior predictive density.

Template Parameters

T_x1	first type of std::vector of scalars
T_x2	second type of std::vector of scalars
T_s	type of parameter sigma, marginal standard deviation
T_l	type of parameter length scale

Parameters

x1	std::vector of Eigen vectors of scalars
x2	std::vector of Eigen vectors of scalars
length_scale	parameter length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x1, x2 are nan or inf

Definition at line 252 of file gp_matern32_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern52_cov ( const std::vector< T_x > &  x, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern 5/2 covariance matrix with one input vector.

\[ k(x, x') = \sigma^2\bigg(1 + \frac{\sqrt{5}d(x, x')}{l} + \frac{5d(x, x')^2}{3l^2}\bigg) exp\bigg(-\frac{5 d(x, x')}{l}\bigg) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x	type of elements contained in vector x
T_s	type of element of sigma, the magnitude
T_l	type of elements of length scale

Parameters

x	std::vector of elements that can be used in stan::math::distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 42 of file gp_matern52_cov.hpp.

template<typename T_x , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern52_cov ( const std::vector< Eigen::Matrix< T_x, Eigen::Dynamic, 1 >> &  x, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern 5/2 covariance matrix with one input vector with automatic relevance determination (ARD).

\[ k(x, x') = \sigma^2\bigg(1 + \sqrt{5}\sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}} + \frac{5}{3} \sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}\bigg) exp\bigg(-\frac{5}{3}\bigg(\sqrt{\sum_{k=1}^K{\frac{d(x, x')^2}{l_k^2}} }\bigg)\bigg) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x	type of elements contained in vector x
T_s	type of element of sigma, the magnitude
T_l	type of elements in vector of length scale

Parameters

x	std::vector of elements that can be used in stan::math::distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain	error if sigma <= 0, l <= 0, or x is nan or inf
std::invalid_argument	if length scale size != dimension of x

Definition at line 107 of file gp_matern52_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern52_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_s &  sigma, const T_l &  length_scale  )

inline

Returns a Matern 5/2 covariance matrix with two different input vectors.

\[ k(x, x') = \sigma^2\bigg(1 + \frac{\sqrt{5}d(x, x')}{l} + \frac{5d(x, x')^2}{3l^2}\bigg) exp\bigg(-\frac{5 d(x, x')}{l}\bigg) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x1	type of elements contained in vector x1
T_x2	type of elements contained in vector x2
T_s	type of element of sigma, the magnitude
T_l	type of elements of length scale

Parameters

x1	std::vector of elements that can be used in stan::math::distance
x2	std::vector of elements that can be used in stan::math::distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain

error if sigma <= 0, l <= 0, or x is nan or inf

Definition at line 175 of file gp_matern52_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_s , typename T_l >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_s, T_l>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_matern52_cov ( const std::vector< Eigen::Matrix< T_x1, Eigen::Dynamic, 1 >> &  x1, const std::vector< Eigen::Matrix< T_x2, Eigen::Dynamic, 1 >> &  x2, const T_s &  sigma, const std::vector< T_l > &  length_scale  )

inline

Returns a Matern 5/2 covariance matrix with two input vectors with automatic relevance determination (ARD).

\[ k(x, x') = \sigma^2\bigg(1 + \sqrt{5}\sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}} + \frac{5}{3} \sqrt{\sum_{k=1}^{K}\frac{d(x, x')^2}{l_k^2}}\bigg) exp\bigg(-\frac{5}{3}\bigg(\sqrt{\sum_{k=1}^K{\frac{d(x, x')^2}{l_k^2}} }\bigg)\bigg) \]

where \( d(x, x') \) is the Euclidean distance.

Template Parameters

T_x1	type of elements contained in vector x1
T_x2	type of elements contained in vector x2
T_s	type of element of sigma, the magnitude
T_l	type of elements in vector of length scale

Parameters

x1	std::vector of elements that can be used in stan::math::distance
x2	std::vector of elements that can be used in stan::math::distance
length_scale	length scale
sigma	standard deviation that can be used in stan::math::square

Exceptions

std::domain	error if sigma <= 0, l <= 0, or x is nan or inf
std::invalid_argument	if length scale size != dimension of x1 or x2

Definition at line 245 of file gp_matern52_cov.hpp.

template<typename T_x , typename T_sigma , typename T_l , typename T_p >

Eigen::Matrix<return_type_t<T_x, T_sigma, T_l, T_p>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_periodic_cov ( const std::vector< T_x > &  x, const T_sigma &  sigma, const T_l &  l, const T_p &  p  )

inline

Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \).

The elements of \( \mathbf{K} \) are defined as \( \mathbf{K}_{ij} = k(\mathbf{X}_i,\mathbf{X}_j), \) where \( \mathbf{X}_i \) is the \(i\)-th row of \( \mathbf{X} \) and
\( k(\mathbf{x},\mathbf{x}^\prime) = \sigma^2 \exp\left(-\frac{2\sin^2(\pi |\mathbf{x}-\mathbf{x}^\prime|/p)}{\ell^2}\right), \)
where \( \sigma^2 \), \( \ell \) and \( p \) are the signal variance, length-scale and period.

Template Parameters

T_x	type of std::vector elements of x. T_x can be a scalar, an Eigen::Vector, or an Eigen::RowVector.
T_sigma	type of sigma
T_l	type of length-scale
T_p	type of period

Parameters

x	std::vector of input elements. This function assumes that all elements of x have the same size.
sigma	standard deviation of the signal
l	length-scale
p	period

Returns

periodic covariance matrix

Exceptions

std::domain_error

if sigma <= 0, l <= 0, p <= 0 or x is nan or infinite

Definition at line 47 of file gp_periodic_cov.hpp.

template<typename T_x1 , typename T_x2 , typename T_sigma , typename T_l , typename T_p >

Eigen::Matrix<return_type_t<T_x1, T_x2, T_sigma, T_l, T_p>, Eigen::Dynamic, Eigen::Dynamic> stan::math::gp_periodic_cov ( const std::vector< T_x1 > &  x1, const std::vector< T_x2 > &  x2, const T_sigma &  sigma, const T_l &  l, const T_p &  p  )

inline

Returns a periodic covariance matrix \( \mathbf{K} \) using inputs \( \mathbf{X}_1 \) and \( \mathbf{X}_2 \).

The elements of \( \mathbf{K} \) are defined as \( \mathbf{K}_{ij} = k(\mathbf{X}_{1_i},\mathbf{X}_{2_j}), \) where \( \mathbf{X}_{1_i} \) and \( \mathbf{X}_{2_j} \) are the \(i\)-th and \(j\)-th rows of \( \mathbf{X}_1 \) and \( \mathbf{X}_2 \) and
\( k(\mathbf{x},\mathbf{x}^\prime) = \sigma^2 \exp\left(-\frac{2\sin^2(\pi |\mathbf{x}-\mathbf{x}^\prime|/p)}{\ell^2}\right), \)
where \( \sigma^2 \), \( \ell \) and \( p \) are the signal variance, length-scale and period.

Template Parameters

T_x1	type of std::vector elements of x1 T_x1 can be a scalar, an Eigen::Vector, or an Eigen::RowVector.
T_x2	type of std::vector elements of x2 T_x2 can be a scalar, an Eigen::Vector, or an Eigen::RowVector.
T_sigma	type of sigma
T_l	type of length-scale
T_p	type of period

Parameters

x1	std::vector of first input elements
x2	std::vector of second input elements. This function assumes that all the elements of x1 and x2 have the same sizes.
sigma	standard deviation of the signal
l	length-scale
p	period

Returns

periodic covariance matrix

Exceptions

std::domain_error

if sigma <= 0, l <= 0, p <= 0 , x1 or x2 is nan or infinite

Definition at line 119 of file gp_periodic_cov.hpp.

template<typename T_x >

std::enable_if< std::is_same<typename scalar_type<T_x>::type, double>::value, Eigen::Matrix<var, Eigen::Dynamic, Eigen::Dynamic> >::type stan::math::gp_periodic_cov ( const std::vector< T_x > &  x, const var &  sigma, const var &  l, const var &  p  )

inline

Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \).

The elements of \( \mathbf{K} \) are defined as \( \mathbf{K}_{ij} = k(\mathbf{X}_i,\mathbf{X}_j), \) where \( \mathbf{X}_i \) is the \(i\)-th row of \( \mathbf{X} \) and
\( k(\mathbf{x},\mathbf{x}^\prime) = \sigma^2 \exp\left(-\frac{2\sin^2(\pi |\mathbf{x}-\mathbf{x}^\prime|/p)}{\ell^2}\right), \)
where \( \sigma^2 \), \( \ell \) and \( p \) are the signal variance, length-scale and period.

Parameters

x	std::vector of input elements. Assumes that all elements of x have the same size.
sigma	standard deviation of the signal
l	length-scale
p	period

Returns

periodic covariance matrix

Exceptions

std::domain_error

if sigma <= 0, l <= 0, p <= 0, or x is nan or infinite

Definition at line 305 of file gp_periodic_cov.hpp.

template<typename T_x >

std::enable_if< std::is_same<typename scalar_type<T_x>::type, double>::value, Eigen::Matrix<var, Eigen::Dynamic, Eigen::Dynamic> >::type stan::math::gp_periodic_cov ( const std::vector< T_x > &  x, double  sigma, const var &  l, const var &  p  )

inline

Returns a periodic covariance matrix \( \mathbf{K} \) using the input \( \mathbf{X} \).

The elements of \( \mathbf{K} \) are defined as \( \mathbf{K}_{ij} = k(\mathbf{X}_i,\mathbf{X}_j), \) where \( \mathbf{X}_i \) is the \(i\)-th row of \( \mathbf{X} \) and
\( k(\mathbf{x},\mathbf{x}^\prime) = \sigma^2 \exp\left(-\frac{2\sin^2(\pi |\mathbf{x}-\mathbf{x}^\prime|/p)}{\ell^2}\right), \)
where \( \sigma^2 \), \( \ell \) and \( p \) are the signal variance, length-scale and period.

Parameters

x	std::vector of input elements. Assumes that all elements of x have the same size.
sigma	standard deviation of the signal
l	length-scale
p	period

Returns

periodic covariance matrix

Exceptions

std::domain_error

if sigma <= 0, l <= 0, p <= 0, or x is nan or infinite

Definition at line 359 of file gp_periodic_cov.hpp.

static void stan::math::grad ( vari *  vi )

static

void stan::math::grad ( var &  v, Eigen::Matrix< var, Eigen::Dynamic, 1 > &  x, Eigen::VectorXd &  g  )

inline

Propagate chain rule to calculate gradients starting from the specified variable.

Resizes the input vector to be the correct size.

The grad() function does not itself recover any memory. use recover_memory() or recover_memory_nested() to recover memory.

Parameters

[in]	v	Value of function being differentiated
[in]	x	Variables being differentiated with respect to
[out]	g	Gradient, d/dx v, evaluated at x.

Definition at line 25 of file grad.hpp.

static void stan::math::grad ( vari *  vi )

static

Compute the gradient for all variables starting from the specified root variable implementation.

Does not recover memory. This chainable variable's adjoint is initialized using the method init_dependent() and then the chain rule is applied working down the stack from this vari and calling each vari's chain() method in turn.

This function computes a nested gradient only going back as far as the last nesting.

This function does not recover any memory from the computation.

Parameters

vi	Variable implementation for root of partial derivative propagation.

Definition at line 30 of file grad.hpp.

template<typename T >

void stan::math::grad_2F1	(	T &	g_a1,
		T &	g_b1,
		const T &	a1,
		const T &	a2,
		const T &	b1,
		const T &	z,
		const T &	precision = 1e-10,
		int	max_steps = 1e5
	)

Gradients of the hypergeometric function, 2F1.

Calculate the gradients of the hypergeometric function (2F1) as the power series stopping when the series converges to within precision or throwing when the function takes max_steps steps.

This power-series representation converges for all gradients under the same conditions as the 2F1 function itself.

Template Parameters

T	type of arguments and result

Parameters

[out]	g_a1	g_a1 pointer to array of six values of type T, result.
[out]	g_b1	g_b1 pointer to array of six values of type T, result.
[in]	a1	a1 see generalized hypergeometric function definition.
[in]	a2	a2 see generalized hypergeometric function definition.
[in]	b1	b1 see generalized hypergeometric function definition.
[in]	z	z see generalized hypergeometric function definition.
[in]	precision	precision of the infinite sum.
[in]	max_steps	number of steps to take.

Definition at line 34 of file grad_2F1.hpp.

template<typename T >

void stan::math::grad_F32	(	T *	g,
		const T &	a1,
		const T &	a2,
		const T &	a3,
		const T &	b1,
		const T &	b2,
		const T &	z,
		const T &	precision = 1e-6,
		int	max_steps = 1e5
	)

Gradients of the hypergeometric function, 3F2.

Calculate the gradients of the hypergeometric function (3F2) as the power series stopping when the series converges to within precision or throwing when the function takes max_steps steps.

This power-series representation converges for all gradients under the same conditions as the 3F2 function itself.

Template Parameters

T	type of arguments and result

Parameters

[out]	g	g pointer to array of six values of type T, result.
[in]	a1	a1 see generalized hypergeometric function definition.
[in]	a2	a2 see generalized hypergeometric function definition.
[in]	a3	a3 see generalized hypergeometric function definition.
[in]	b1	b1 see generalized hypergeometric function definition.
[in]	b2	b2 see generalized hypergeometric function definition.
[in]	z	z see generalized hypergeometric function definition.
[in]	precision	precision of the infinite sum
[in]	max_steps	number of steps to take

Definition at line 36 of file grad_F32.hpp.

template<typename F >

void stan::math::grad_hessian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	H,
		std::vector< Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > > &	grad_H
	)

Calculate the value, the Hessian, and the gradient of the Hessian of the specified function at the specified argument.

The functor must implement

fvar<fvar<var> > operator()(const Eigen::Matrix<fvar<fvar<var> >, Eigen::Dynamic, 1>&)

using only operations that are defined for fvar and var.

This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	H	Hessian of function at argument
[out]	grad_H	Gradient of the Hessian of function at argument

Definition at line 42 of file grad_hessian.hpp.

void stan::math::grad_inc_beta ( double &  g1, double &  g2, double  a, double  b, double  z  )

inline

Definition at line 20 of file grad_inc_beta.hpp.

template<typename T >

void stan::math::grad_inc_beta	(	fvar< T > &	g1,
		fvar< T > &	g2,
		fvar< T >	a,
		fvar< T >	b,
		fvar< T >	z
	)

Gradient of the incomplete beta function beta(a, b, z) with respect to the first two arguments.

Uses the equivalence to a hypergeometric function. See http://dlmf.nist.gov/8.17#ii

Template Parameters

T	type of fvar

Parameters

[out]	g1	d/da
[out]	g2	d/db
[in]	a	a
[in]	b	b
[in]	z	z

Definition at line 34 of file grad_inc_beta.hpp.

void stan::math::grad_inc_beta ( var &  g1, var &  g2, const var &  a, const var &  b, const var &  z  )

inline

Gradient of the incomplete beta function beta(a, b, z) with respect to the first two arguments.

Uses the equivalence to a hypergeometric function. See http://dlmf.nist.gov/8.17#ii

Parameters

[out]	g1	d/da
[out]	g2	d/db
[in]	a	a
[in]	b	b
[in]	z	z

Definition at line 36 of file grad_inc_beta.hpp.

template<typename T >

void stan::math::grad_reg_inc_beta	(	T &	g1,
		T &	g2,
		const T &	a,
		const T &	b,
		const T &	z,
		const T &	digammaA,
		const T &	digammaB,
		const T &	digammaSum,
		const T &	betaAB
	)

Computes the gradients of the regularized incomplete beta function.

Specifically, this function computes gradients of ibeta(a, b, z), with respect to the arguments a and b.

Template Parameters

T	type of arguments

Parameters

[out]	g1	partial derivative of ibeta(a, b, z) with respect to a
[out]	g2	partial derivative of ibeta(a, b, z) with respect to b
[in]	a	a
[in]	b	b
[in]	z	z
[in]	digammaA	the value of digamma(a)
[in]	digammaB	the value of digamma(b)
[in]	digammaSum	the value of digamma(a + b)
[in]	betaAB	the value of beta(a, b)

Definition at line 33 of file grad_reg_inc_beta.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::grad_reg_inc_gamma	(	T1	a,
		T2	z,
		T1	g,
		T1	dig,
		double	precision = 1e-6,
		int	max_steps = 1e5
	)

Gradient of the regularized incomplete gamma functions igamma(a, z)

For small z, the gradient is computed via the series expansion; for large z, the series is numerically inaccurate due to cancellation and the asymptotic expansion is used.

Parameters

a	shape parameter, a > 0
z	location z >= 0
g	stan::math::tgamma(a) (precomputed value)
dig	boost::math::digamma(a) (precomputed value)
precision	required precision; applies to series expansion only
max_steps	number of steps to take.

Exceptions

throws

std::domain_error if not converged after max_steps or increment overflows to inf.

For the asymptotic expansion, the gradient is given by:

\[ \begin{array}{rcl} \Gamma(a, z) & = & z^{a-1}e^{-z} \sum_{k=0}^N \frac{(a-1)_k}{z^k} \qquad , z \gg a\\ Q(a, z) & = & \frac{z^{a-1}e^{-z}}{\Gamma(a)} \sum_{k=0}^N \frac{(a-1)_k}{z^k}\\ (a)_k & = & (a)_{k-1}(a-k)\\ \frac{d}{da} (a)_k & = & (a)_{k-1} + (a-k)\frac{d}{da} (a)_{k-1}\\ \frac{d}{da}Q(a, z) & = & (log(z) - \psi(a)) Q(a, z)\\ && + \frac{z^{a-1}e^{-z}}{\Gamma(a)} \sum_{k=0}^N \left(\frac{d}{da} (a-1)_k\right) \frac{1}{z^k} \end{array} \]

Definition at line 49 of file grad_reg_inc_gamma.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::grad_reg_lower_inc_gamma	(	const T1 &	a,
		const T2 &	z,
		double	precision = 1e-10,
		int	max_steps = 1e5
	)

Computes the gradient of the lower regularized incomplete gamma function.

The lower incomlete gamma function derivative w.r.t it's first parameter (a) seems to have no standard source. It also appears to have no widely known approximate implementation. Gautschi (1979) has a thorough discussion of the calculation of the lower regularized incomplete gamma function itself and some stability issues.

Reference: Gautschi, Walter (1979) ACM Transactions on mathematical software. 5(4):466-481

We implemented calculations for d(gamma_p)/da by taking derivatives of formulas suggested by Gauschi and others and testing them against an outside source (Mathematica). We took three implementations which can cover the range {a:[0,20], z:[0,30]} with absolute error < 1e-10 with the exception of values near (0,0) where the error is near 1e-5. Relative error is also <<1e-6 except for regions where the gradient approaches zero.

Gautschi suggests calculating the lower incomplete gamma function for small to moderate values of $z$ using the approximation:

\[ \frac{\gamma(a,z)}{\Gamma(a)}=z^a e^-z \sum_n=0^\infty \frac{z^n}{\Gamma(a+n+1)} \]

We write the derivative in the form:

\[ \frac{d\gamma(a,z)\Gamma(a)}{da} = \frac{\log z}{e^z} \sum_n=0^\infty \frac{z^{a+n}}{\Gamma(a+n+1)} - \frac{1}{e^z} \sum_n=0^\infty \frac{z^{a+n}}{\Gamma(a+n+1)}\psi^0(a+n+1) \]

This caclulation is sufficiently accurate for small $a$ and small $z$. For larger values and $a$ and $z$ we use it in its log form:

\[ \frac{d \gamma(a,z)\Gamma(a)}{da} = \frac{\log z}{e^z} \sum_n=0^\infty \exp[(a+n)\log z - \log\Gamma(a+n+1)] - \sum_n=0^\infty \exp[(a+n)\log z - \log\Gamma(a+n+1) + \log\psi^0(a+n+1)] \]

For large $z$, Gauschi recommends using the upper incomplete Gamma instead and the negative of its derivative turns out to be more stable and accurate for larger $z$ and for some combinations of $a$ and $z$. This is a log-scale implementation of the derivative of the formulation suggested by Gauschi (1979). For some values it defers to the negative of the gradient for the gamma_q function. This is based on the suggestion by Gauschi (1979) that for large values of $z$ it is better to carry out calculations using the upper incomplete Gamma function.

Branching for choice of implementation for the lower incomplete regularized gamma function gradient. The derivative based on Gautschi's formulation appears to be sufficiently accurate everywhere except for large z and small to moderate a. The intersection between the two regions is a radius 12 quarter circle centered at a=0, z=30 although both implementations are satisfactory near the intersection.

Some limits that could be treated, e.g., infinite z should return tgamma(a) * digamma(a), throw instead to match the behavior of, e.g., boost::math::gamma_p

Template Parameters

T1	type of a
T2	type of z

Parameters

[in]	a	shared with complete Gamma
[in]	z	value to integrate up to
[in]	precision	series terminates when increment falls below this value.
[in]	max_steps	number of terms to sum before throwing

Exceptions

std::domain_error

if the series does not converge to requested precision before max_steps.

Definition at line 108 of file grad_reg_lower_inc_gamma.hpp.

template<typename F >

void stan::math::grad_tr_mat_times_hessian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	M,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad_tr_MH
	)

Definition at line 15 of file grad_tr_mat_times_hessian.hpp.

template<typename T , typename F >

void stan::math::gradient	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	grad_fx
	)

Calculate the value and the gradient of the specified function at the specified argument.

The functor must implement

fvar<T> operator()(const Eigen::Matrix<T, Eigen::Dynamic, 1>&)

using only operations that are defined for fvar. This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Time and memory usage is on the order of the size of the fully unfolded expression for the function applied to the argument, independently of dimension.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument

Definition at line 39 of file gradient.hpp.

template<typename F >

void stan::math::gradient	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad_fx
	)

Calculate the value and the gradient of the specified function at the specified argument.

The functor must implement

var operator()(const Eigen::Matrix<var, Eigen::Dynamic, 1>&)

using only operations that are defined for var. This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Time and memory usage is on the order of the size of the fully unfolded expression for the function applied to the argument, independently of dimension.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument

Definition at line 42 of file gradient.hpp.

template<typename T1 , typename T2 , typename F >

void stan::math::gradient_dot_vector	(	const F &	f,
		const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &	v,
		T1 &	fx,
		T1 &	grad_fx_dot_v
	)

Definition at line 13 of file gradient_dot_vector.hpp.

template<typename F >

double stan::math::gradient_of_f ( const F &  f, const double &  x, const double &  xc, const std::vector< double > &  theta_vals, const std::vector< double > &  x_r, const std::vector< int > &  x_i, size_t  n, std::ostream *  msgs  )

inline

Calculate first derivative of f(x, param, std::ostream&) with respect to the nth parameter.

Uses nested reverse mode autodiff

Gradients that evaluate to NaN are set to zero if the function itself evaluates to zero. If the function is not zero and the gradient evaluates to NaN, a std::domain_error is thrown

Definition at line 31 of file integrate_1d.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::gumbel_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  beta  )

inline

Definition at line 100 of file gumbel_lpdf.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::head ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v, size_t  n  )

inline

Return the specified number of elements as a vector from the front of the specified vector.

Template Parameters

T	type of elements in the vector

Parameters

v	Vector input.
n	Size of return.

Returns

The first n elements of v.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 22 of file head.hpp.

template<typename T >

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::head ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  rv, size_t  n  )

inline

Return the specified number of elements as a row vector from the front of the specified row vector.

Template Parameters

T	type of elements in the vector

Parameters

rv	Row vector.
n	Size of return row vector.

Returns

The first n elements of rv.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 41 of file head.hpp.

template<typename T >

std::vector<T> stan::math::head	(	const std::vector< T > &	sv,
		size_t	n
	)

Return the specified number of elements as a standard vector from the front of the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

sv	Standard vector.
n	Size of return.

Returns

The first n elements of sv.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 60 of file head.hpp.

template<typename T , typename F >

void stan::math::hessian	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	grad,
		Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	H
	)

Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in time O(N^3) time and O(N^2) space.

The advantage over the mixed definition, which is faster for Hessians, is that this version is itself differentiable.

The functor must implement

fvar<fvar<T> > operator()(const Eigen::Matrix<fvar<fvar<T> >, Eigen::Dynamic, 1>&)

using only operations that are defined for the argument type.

This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Template Parameters

T	Type of underlying scalar
F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad	gradient of function at argument
[out]	H	Hessian of function at argument

Definition at line 41 of file hessian.hpp.

template<typename F >

void stan::math::hessian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad,
		Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	H
	)

Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in O(N^2) time and O(N^2) space.

The functor must implement

fvar<var> operator()(const Eigen::Matrix<fvar<var>, Eigen::Dynamic, 1>&)

using only operations that are defined for fvar and var.

This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad	gradient of function at argument
[out]	H	Hessian of function at argument

Definition at line 42 of file hessian.hpp.

template<typename F >

void stan::math::hessian_times_vector	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	v,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	Hv
	)

Definition at line 14 of file hessian_times_vector.hpp.

template<typename T , typename F >

void stan::math::hessian_times_vector	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	v,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	Hv
	)

Definition at line 42 of file hessian_times_vector.hpp.

template<bool propto, typename T_n , typename T_N , typename T_a , typename T_b >

double stan::math::hypergeometric_lpmf	(	const T_n &	n,
		const T_N &	N,
		const T_a &	a,
		const T_b &	b
	)

Definition at line 16 of file hypergeometric_lpmf.hpp.

template<typename T_n , typename T_N , typename T_a , typename T_b >

double stan::math::hypergeometric_lpmf ( const T_n &  n, const T_N &  N, const T_a &  a, const T_b &  b  )

inline

Definition at line 56 of file hypergeometric_lpmf.hpp.

template<class RNG >

int stan::math::hypergeometric_rng ( int  N, int  a, int  b, RNG &  rng  )

inline

Definition at line 14 of file hypergeometric_rng.hpp.

template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

double stan::math::hypot ( T1  x, T2  y  )

inline

Return the length of the hypoteneuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11).

In symbols, if the arguments are x and y, the result is sqrt(x * x + y * y).

Parameters

x	First argument.
y	Second argument.

Returns

Length of hypoteneuse of right triangle with opposite and adjacent side lengths x and y.

Definition at line 23 of file hypot.hpp.

template<typename T >

fvar<T> stan::math::hypot ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the length of the hypoteneuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11).

In symbols, if the arguments are 1 and x2, the result is sqrt(x1 * x1 + x2 * x2).

Template Parameters

T	Scalar type of autodiff variables.

Parameters

x1	First argument.
x2	Second argument.

Returns

Length of hypoteneuse of right triangle with opposite and adjacent side lengths x1 and x2.

Definition at line 26 of file hypot.hpp.

var stan::math::hypot ( const var &  a, const var &  b  )

inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

The partial derivatives are given by

\(\frac{\partial}{\partial x} \sqrt{x^2 + y^2} = \frac{x}{\sqrt{x^2 + y^2}}\), and

\(\frac{\partial}{\partial y} \sqrt{x^2 + y^2} = \frac{y}{\sqrt{x^2 + y^2}}\).

Parameters

[in]	a	Length of first side.
[in]	b	Length of second side.

Returns

Length of hypoteneuse.

Definition at line 45 of file hypot.hpp.

template<typename T >

fvar<T> stan::math::hypot ( const fvar< T > &  x1, double  x2  )

inline

Return the length of the hypoteneuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11).

In symbols, if the arguments are 1 and x2, the result is sqrt(x1 * x1 + x2 * x2).

Template Parameters

T	Scalar type of autodiff variable.

Parameters

x1	First argument.
x2	Second argument.

Returns

Length of hypoteneuse of right triangle with opposite and adjacent side lengths x1 and x2.

Definition at line 46 of file hypot.hpp.

var stan::math::hypot ( const var &  a, double  b  )

inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

The derivative is

\(\frac{d}{d x} \sqrt{x^2 + c^2} = \frac{x}{\sqrt{x^2 + c^2}}\).

Parameters

[in]	a	Length of first side.
[in]	b	Length of second side.

Returns

Length of hypoteneuse.

Definition at line 61 of file hypot.hpp.

template<typename T >

fvar<T> stan::math::hypot ( double  x1, const fvar< T > &  x2  )

inline

Return the length of the hypoteneuse of a right triangle with opposite and adjacent side lengths given by the specified arguments (C++11).

In symbols, if the arguments are 1 and x2, the result is sqrt(x1 * x1 + x2 * x2).

Template Parameters

T	Scalar type of autodiff variable.

Parameters

x1	First argument.
x2	Second argument.

Returns

Length of hypoteneuse of right triangle with opposite and adjacent side lengths x1 and x2.

Definition at line 66 of file hypot.hpp.

var stan::math::hypot ( double  a, const var &  b  )

inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

The derivative is

\(\frac{d}{d y} \sqrt{c^2 + y^2} = \frac{y}{\sqrt{c^2 + y^2}}\).

\[ \mbox{hypot}(x, y) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \sqrt{x^2+y^2} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{hypot}(x, y)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \frac{x}{\sqrt{x^2+y^2}} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{hypot}(x, y)}{\partial y} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \frac{y}{\sqrt{x^2+y^2}} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

[in]	a	Length of first side.
[in]	b	Length of second side.

Returns

Length of hypoteneuse.

Definition at line 104 of file hypot.hpp.

double stan::math::ibeta ( double  a, double  b, double  x  )

inline

The normalized incomplete beta function of a, b, and x.

Used to compute the cumulative density function for the beta distribution.

Parameters

a	Shape parameter a <= 0; a and b can't both be 0
b	Shape parameter b <= 0
x	Random variate. 0 <= x <= 1

Exceptions

if	constraints are violated or if any argument is NaN

Returns

The normalized incomplete beta function.

Definition at line 24 of file ibeta.hpp.

var stan::math::ibeta ( const var &  a, const var &  b, const var &  x  )

inline

The normalized incomplete beta function of a, b, and x.

Used to compute the cumulative density function for the beta distribution.

Partial derivatives are those specified by wolfram alpha. The values were checked using both finite differences and by independent code for calculating the derivatives found in JSS (paper by Boik and Robison-Cox).

Parameters

a	Shape parameter.
b	Shape parameter.
x	Random variate.

Returns

The normalized incomplete beta function.

Exceptions

if	any argument is NaN.

Definition at line 204 of file ibeta.hpp.

template<typename T >

T stan::math::identity_constrain ( const T &  x )

inline

Returns the result of applying the identity constraint transform to the input.

This method is effectively a no-op and is mainly useful as a placeholder in auto-generated code.

Template Parameters

T	Type of scalar.

Parameters

[in]

x

free scalar

Returns

transformed input

Definition at line 20 of file identity_constrain.hpp.

template<typename T >

T stan::math::identity_constrain ( const T &  x, T &    )

inline

Returns the result of applying the identity constraint transform to the input and increments the log probability reference with the log absolute Jacobian determinant.

This method is effectively a no-op and mainly useful as a placeholder in auto-generated code.

Template Parameters

T	type of scalar.

Parameters

[in]

x

scalar

Returns

transformed input

Definition at line 37 of file identity_constrain.hpp.

template<typename T >

T stan::math::identity_free ( const T &  y )

inline

Returns the result of applying the inverse of the identity constraint transform to the input.

This function is a no-op and mainly useful as a placeholder in auto-generated code.

Template Parameters

T	type of value

Parameters

[in]

y

value

Returns

value

Definition at line 20 of file identity_free.hpp.

var stan::math::if_else ( bool  c, const var &  y_true, const var &  y_false  )

inline

If the specified condition is true, return the first variable, otherwise return the second variable.

Parameters

c	Boolean condition.
y_true	Variable to return if condition is true.
y_false	Variable to return if condition is false.

Definition at line 18 of file if_else.hpp.

template<typename T_true , typename T_false >

return_type_t<T_true, T_false> stan::math::if_else ( const bool  c, const T_true  y_true, const T_false  y_false  )

inline

Return the second argument if the first argument is true and otherwise return the second argument.

This is just a convenience method to provide a function with the same behavior as the built-in ternary operator. In general, this function behaves as if defined by

if_else(c, y1, y0) = c ? y1 : y0.

Parameters

c	Boolean condition value.
y_true	Value to return if condition is true.
y_false	Value to return if condition is false.

Definition at line 24 of file if_else.hpp.

var stan::math::if_else ( bool  c, double  y_true, const var &  y_false  )

inline

If the specified condition is true, return a new variable constructed from the first scalar, otherwise return the second variable.

Parameters

c	Boolean condition.
y_true	Value to promote to variable and return if condition is true.
y_false	Variable to return if condition is false.

Definition at line 30 of file if_else.hpp.

var stan::math::if_else ( bool  c, const var &  y_true, double  y_false  )

inline

If the specified condition is true, return the first variable, otherwise return a new variable constructed from the second scalar.

Parameters

c	Boolean condition.
y_true	Variable to return if condition is true.
y_false	Value to promote to variable and return if condition is false.

Definition at line 46 of file if_else.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( const fvar< T > &  a, const fvar< T > &  b, const fvar< T > &  x  )

inline

Definition at line 19 of file inc_beta.hpp.

double stan::math::inc_beta ( double  a, double  b, double  x  )

inline

The normalized incomplete beta function of a, b, with outcome x.

Used to compute the cumulative density function for the beta distribution.

Parameters

a	Shape parameter a <= 0; a and b can't both be 0
b	Shape parameter b <= 0
x	Random variate. 0 <= x <= 1

Exceptions

if	constraints are violated or if any argument is NaN

Returns

The normalized incomplete beta function.

Definition at line 24 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( double  a, const fvar< T > &  b, const fvar< T > &  x  )

inline

Definition at line 32 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( const fvar< T > &  a, double  b, const fvar< T > &  x  )

inline

Definition at line 36 of file inc_beta.hpp.

var stan::math::inc_beta ( const var &  a, const var &  b, const var &  c  )

inline

Definition at line 38 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( const fvar< T > &  a, const fvar< T > &  b, double  x  )

inline

Definition at line 40 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( double  a, double  b, const fvar< T > &  x  )

inline

Definition at line 45 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( const fvar< T > &  a, double  b, double  x  )

inline

Definition at line 49 of file inc_beta.hpp.

template<typename T >

fvar<T> stan::math::inc_beta ( double  a, const fvar< T > &  b, double  x  )

inline

Definition at line 53 of file inc_beta.hpp.

template<typename T >

T stan::math::inc_beta_dda	(	T	a,
		T	b,
		T	z,
		T	digamma_a,
		T	digamma_ab
	)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to a.

The power series used to compute the deriative tends to converge slowly when a and b are large, especially if z approaches 1. The implementation will throw an exception if the series have not converged within 100,000 iterations. The current implementation has been tested for values of a and b up to 12500 and z = 0.999.

Template Parameters

T	scalar types of arguments

Parameters

a	a
b	b
z	upper bound of the integral
digamma_a	value of digamma(a)
digamma_ab	value of digamma(b)

Returns

partial derivative of the incomplete beta with respect to a

Precondition

a >= 0

b >= 0

0 <= z <= 1

Definition at line 36 of file inc_beta_dda.hpp.

template<typename T >

T stan::math::inc_beta_ddb	(	T	a,
		T	b,
		T	z,
		T	digamma_b,
		T	digamma_ab
	)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to b.

The power series used to compute the deriative tends to converge slowly when a and b are large, especailly if z approaches 1. The implementation will throw an exception if the series have not converged within 100,000 iterations. The current implementation has been tested for values of a and b up to 12500 and z = 0.999.

Template Parameters

T	scalar types of arguments

Parameters

a	a
b	b
z	upper bound of the integral
digamma_b	value of digamma(b)
digamma_ab	value of digamma(b)

Returns

partial derivative of the incomplete beta with respect to b

Precondition

a >= 0

b >= 0

0 <= z <= 1

Definition at line 39 of file inc_beta_ddb.hpp.

template<typename T >

T stan::math::inc_beta_ddz	(	T	a,
		T	b,
		T	z
	)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to z.

Template Parameters

T	scalar types of arguments

Parameters

a	a
b	b
z	upper bound of the integral

Returns

partial derivative of the incomplete beta with respect to z

Precondition

a > 0

b > 0

0 < z <= 1

Definition at line 27 of file inc_beta_ddz.hpp.

template<>

double stan::math::inc_beta_ddz ( double  a, double  b, double  z  )

inline

Definition at line 35 of file inc_beta_ddz.hpp.

template<std::size_t N, class F >

constexpr auto stan::math::index_apply

(

F

f

)

Calls given callable with an index sequence.

Combination of this and a lambda can be used to avoid separating implementation that needs an index sequence from functions that operate on tuples of parameter packs.

Template Parameters

N	size of index sequence to use
F	callable type

Parameters

f

callable

Returns

what the given callable returns

Definition at line 26 of file index_apply.hpp.

tbb::task_scheduler_init& stan::math::init_threadpool_tbb ( tbb::stack_size_type  stack_size = 0 )

inline

Initialize the Intel TBB threadpool and global scheduler through the tbb::task_scheduler_init object.

In case an instance of the tbb::task_scheduler_object has been instantiated prior to calling this function, then any subsequent initialization is ignored by the Intel TBB.

The maximal number of threads is read from the environment variable STAN_NUM_THREADS using internal::get_num_threads. See conventions of get_num_threads. The TBB scheduler will be activated by calling this function.

The function returns a reference to the static tbb::task_scheduler_init instance.

Parameters

stack_size

sets the stack size of each thread; the default 0 let's the TBB choose the stack size

Returns

reference to the static tbb::task_scheduler_init

Exceptions

std::runtime_error

if the value of STAN_NUM_THREADS env. variable is invalid

Definition at line 83 of file init_threadpool_tbb.hpp.

template<typename T >

void stan::math::initialize ( T &  x, const T &  v  )

inline

Definition at line 16 of file initialize.hpp.

template<typename T , typename V , typename = require_arithmetic_t<V>>

void stan::math::initialize ( T &  x, V  v  )

inline

Definition at line 20 of file initialize.hpp.

template<typename T , int R, int C, typename V >

void stan::math::initialize ( Eigen::Matrix< T, R, C > &  x, const V &  v  )

inline

Definition at line 24 of file initialize.hpp.

template<typename T , typename V >

void stan::math::initialize ( std::vector< T > &  x, const V &  v  )

inline

Definition at line 30 of file initialize.hpp.

void stan::math::initialize_variable ( var &  variable, const var &  value  )

inline

Initialize variable to value.

(Function may look pointless, but its needed to bottom out recursion.)

Definition at line 16 of file initialize_variable.hpp.

template<int R, int C>

void stan::math::initialize_variable ( Eigen::Matrix< var, R, C > &  matrix, const var &  value  )

inline

Initialize every cell in the matrix to the specified value.

Definition at line 25 of file initialize_variable.hpp.

template<typename T >

void stan::math::initialize_variable ( std::vector< T > &  variables, const var &  value  )

inline

Initialize the variables in the standard vector recursively.

Definition at line 34 of file initialize_variable.hpp.

template<typename T >

int stan::math::int_step ( const T &  y )

inline

The integer step, or Heaviside, function.

For double NaN input, int_step(NaN) returns 0.

\[ \mbox{int\_step}(x) = \begin{cases} 0 & \mbox{if } x \leq 0 \\ 1 & \mbox{if } x > 0 \\[6pt] 0 & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters

T	value type

Parameters

[in]

y

value

Returns

1 if value is greater than 0 and 0 otherwise

Definition at line 27 of file int_step.hpp.

template<typename F >

double stan::math::integrate ( const F &  f, double  a, double  b, double  relative_tolerance  )

inline

Integrate a single variable function f from a to b to within a specified relative tolerance.

This function assumes a is less than b.

The signature for f should be: double f(double x, double xc)

It should return the value of the function evaluated at x.

Depending on whether or not a is finite or negative infinity and b is finite or positive infinity, a different version of the 1d quadrature algorithm from the Boost quadrature library is chosen.

Integrals that cross zero are broken into two, and the separate integrals are each integrated to the given relative tolerance.

For integrals with finite limits, the xc argument is the distance to the nearest boundary. So for a > 0, b > 0, it will be a - x for x closer to a, and b - x for x closer to b. xc is computed in a way that avoids the precision loss of computing a - x or b - x manually. For integrals that cross zero, xc can take values a - x, -x, or b - x depending on which integration limit it is nearest.

If either limit is infinite, xc is set to NaN

Template Parameters

T

Type of f

Parameters

f	the function to be integrated
a	lower limit of integration
b	upper limit of integration
relative_tolerance	target relative tolerance passed to Boost quadrature

Returns

numeric integral of function f

If the integral crosses zero, break it into two (advice from the Boost implementation: https://www.boost.org/doc/libs/1_66_0/libs/math/doc/html/math_toolkit/double_exponential/de_caveats.html)

Definition at line 51 of file integrate_1d.hpp.

template<typename F , typename T_a , typename T_b , typename T_theta , typename = require_any_var_t<T_a, T_b, T_theta>>

return_type_t<T_a, T_b, T_theta> stan::math::integrate_1d ( const F &  f, const T_a &  a, const T_b &  b, const std::vector< T_theta > &  theta, const std::vector< double > &  x_r, const std::vector< int > &  x_i, std::ostream *  msgs, const double  relative_tolerance = std::sqrt(EPSILON)  )

inline

Compute the integral of the single variable function f from a to b to within a specified relative tolerance.

a and b can be finite or infinite.

f should be compatible with reverse mode autodiff and have the signature: var f(double x, double xc, const std::vector<var>& theta, const std::vector<double>& x_r, const std::vector<int> &x_i, std::ostream* msgs)

It should return the value of the function evaluated at x. Any errors should be printed to the msgs stream.

Integrals that cross zero are broken into two, and the separate integrals are each integrated to the given relative tolerance.

For integrals with finite limits, the xc argument is the distance to the nearest boundary. So for a > 0, b > 0, it will be a - x for x closer to a, and b - x for x closer to b. xc is computed in a way that avoids the precision loss of computing a - x or b - x manually. For integrals that cross zero, xc can take values a - x, -x, or b - x depending on which integration limit it is nearest.

If either limit is infinite, xc is set to NaN

The integration algorithm terminates when

\[ \frac{{|I_{n + 1} - I_n|}}{{|I|_{n + 1}}} < \text{relative tolerance} \]

where \(I_{n}\) is the nth estimate of the integral and \(|I|_{n}\) is the nth estimate of the norm of the integral.

Integrals that cross zero are split into two. In this case, each integral is separately integrated to the given relative_tolerance.

Gradients of f that evaluate to NaN when the function evaluates to zero are set to zero themselves. This is due to the autodiff easily overflowing to NaN when evaluating gradients near the maximum and minimum floating point values (where the function should be zero anyway for the integral to exist)

Template Parameters

T_a	type of first limit
T_b	type of second limit
T_theta	type of parameters
T	Type of f

Parameters

	f	the functor to integrate
	a	lower limit of integration
	b	upper limit of integration
	theta	additional parameters to be passed to f
	x_r	additional data to be passed to f
	x_i	additional integer data to be passed to f
[in,out]	msgs	the print stream for warning messages
	relative_tolerance	relative tolerance passed to Boost quadrature

Returns

numeric integral of function f

Definition at line 119 of file integrate_1d.hpp.

template<typename F >

double stan::math::integrate_1d ( const F &  f, const double  a, const double  b, const std::vector< double > &  theta, const std::vector< double > &  x_r, const std::vector< int > &  x_i, std::ostream *  msgs, const double  relative_tolerance = std::sqrt(EPSILON)  )

inline

Compute the integral of the single variable function f from a to b to within a specified relative tolerance.

a and b can be finite or infinite.

The signature for f should be: double f(double x, double xc, const std::vector<double>& theta, const std::vector<double>& x_r, const std::vector<int>& x_i, std::ostream* msgs)

It should return the value of the function evaluated at x. Any errors should be printed to the msgs stream.

Integrals that cross zero are broken into two, and the separate integrals are each integrated to the given relative tolerance.

For integrals with finite limits, the xc argument is the distance to the nearest boundary. So for a > 0, b > 0, it will be a - x for x closer to a, and b - x for x closer to b. xc is computed in a way that avoids the precision loss of computing a - x or b - x manually. For integrals that cross zero, xc can take values a - x, -x, or b - x depending on which integration limit it is nearest.

If either limit is infinite, xc is set to NaN

The integration algorithm terminates when

\[ \frac{{|I_{n + 1} - I_n|}}{{|I|_{n + 1}}} < \text{relative tolerance} \]

where \(I_{n}\) is the nth estimate of the integral and \(|I|_{n}\) is the nth estimate of the norm of the integral.

Integrals that cross zero are split into two. In this case, each integral is separately integrated to the given relative_tolerance.

Template Parameters

T

Type of f

Parameters

	f	the function to be integrated
	a	lower limit of integration
	b	upper limit of integration
	theta	additional parameters to be passed to f
	x_r	additional data to be passed to f
	x_i	additional integer data to be passed to f
[in,out]	msgs	the print stream for warning messages
	relative_tolerance	tolerance passed to Boost quadrature

Returns

numeric integral of function f

Definition at line 185 of file integrate_1d.hpp.

template<typename F , typename Tpar >

std::vector<std::vector<Tpar> > stan::math::integrate_dae	(	const F &	f,
		const std::vector< double > &	yy0,
		const std::vector< double > &	yp0,
		double	t0,
		const std::vector< double > &	ts,
		const std::vector< Tpar > &	theta,
		const std::vector< double > &	x_r,
		const std::vector< int > &	x_i,
		const double	rtol,
		const double	atol,
		const int64_t	max_num_steps = idas_integrator::IDAS_MAX_STEPS,
		std::ostream *	msgs = nullptr
	)

Return the solutions for a semi-explicit DAE system with residual specified by functor F, given the specified consistent initial state yy0 and yp0.

Template Parameters

DAE	type of DAE system
Tpar	scalar type of parameter theta

Parameters

[in]	f	functor for the base ordinary differential equation
[in]	yy0	initial state
[in]	yp0	initial derivative state
[in]	t0	initial time
[in]	ts	times of the desired solutions, in strictly increasing order, all greater than the initial time
[in]	theta	parameters
[in]	x_r	real data
[in]	x_i	int data
[in]	rtol	relative tolerance passed to IDAS, requred <10^-3
[in]	atol	absolute tolerance passed to IDAS, problem-dependent
[in]	max_num_steps	maximal number of admissable steps between time-points
[in]	msgs	message

Returns

a vector of states, each state being a vector of the same size as the state variable, corresponding to a time in ts.

Definition at line 37 of file integrate_dae.hpp.

template<typename F , typename T_initial , typename T_param , typename T_t0 , typename T_ts >

std::vector<std::vector< typename stan::return_type<T_initial, T_param, T_t0, T_ts>::type> > stan::math::integrate_ode_adams	(	const F &	f,
		const std::vector< T_initial > &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		const std::vector< T_param > &	theta,
		const std::vector< double > &	x,
		const std::vector< int > &	x_int,
		std::ostream *	msgs = nullptr,
		double	relative_tolerance = 1e-10,
		double	absolute_tolerance = 1e-10,
		long int	max_num_steps = 1e8
	)

Definition at line 16 of file integrate_ode_adams.hpp.

template<typename F , typename T_initial , typename T_param , typename T_t0 , typename T_ts >

std::vector<std::vector< typename stan::return_type<T_initial, T_param, T_t0, T_ts>::type> > stan::math::integrate_ode_bdf	(	const F &	f,
		const std::vector< T_initial > &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		const std::vector< T_param > &	theta,
		const std::vector< double > &	x,
		const std::vector< int > &	x_int,
		std::ostream *	msgs = nullptr,
		double	relative_tolerance = 1e-10,
		double	absolute_tolerance = 1e-10,
		long int	max_num_steps = 1e8
	)

Definition at line 16 of file integrate_ode_bdf.hpp.

template<typename F , typename T1 , typename T2 , typename T_t0 , typename T_ts >

std::vector<std::vector<return_type_t<T1, T2, T_t0, T_ts> > > stan::math::integrate_ode_rk45	(	const F &	f,
		const std::vector< T1 > &	y0,
		const T_t0 &	t0,
		const std::vector< T_ts > &	ts,
		const std::vector< T2 > &	theta,
		const std::vector< double > &	x,
		const std::vector< int > &	x_int,
		std::ostream *	msgs = nullptr,
		double	relative_tolerance = 1e-6,
		double	absolute_tolerance = 1e-6,
		int	max_num_steps = 1E6
	)

Return the solutions for the specified system of ordinary differential equations given the specified initial state, initial times, times of desired solution, and parameters and data, writing error and warning messages to the specified stream.

Warning: If the system of equations is stiff, roughly defined by having varying time scales across dimensions, then this solver is likely to be slow.

This function is templated to allow the initial times to be either data or autodiff variables and the parameters to be data or autodiff variables. The autodiff-based implementation for reverse-mode are defined in namespace stan::math and may be invoked via argument-dependent lookup by including their headers.

This function uses the Dormand-Prince method as implemented in Boost's boost::numeric::odeint::runge_kutta_dopri5 integrator.

Template Parameters

F	type of ODE system function.
T1	type of scalars for initial values.
T2	type of scalars for parameters.
T_t0	type of scalar of initial time point.
T_ts	type of time-points where ODE solution is returned.

Parameters

[in]	f	functor for the base ordinary differential equation.
[in]	y0	initial state.
[in]	t0	initial time.
[in]	ts	times of the desired solutions, in strictly increasing order, all greater than the initial time.
[in]	theta	parameter vector for the ODE.
[in]	x	continuous data vector for the ODE.
[in]	x_int	integer data vector for the ODE.
[out]	msgs	the print stream for warning messages.
[in]	relative_tolerance	relative tolerance parameter for Boost's ode solver. Defaults to 1e-6.
[in]	absolute_tolerance	absolute tolerance parameter for Boost's ode solver. Defaults to 1e-6.
[in]	max_num_steps	maximum number of steps to take within the Boost ode solver.

Returns

a vector of states, each state being a vector of the same size as the state variable, corresponding to a time in ts.

Definition at line 70 of file integrate_ode_rk45.hpp.

double stan::math::inv ( double  x )

inline

Definition at line 8 of file inv.hpp.

template<typename T >

fvar<T> stan::math::inv ( const fvar< T > &  x )

inline

Definition at line 12 of file inv.hpp.

template<typename T >

apply_scalar_unary<inv_fun, T>::return_t stan::math::inv ( const T &  x )

inline

Vectorized version of inv().

Template Parameters

T	type of container

Parameters

x

container

Returns

1 divided by each value in x.

Definition at line 32 of file inv.hpp.

var stan::math::inv ( const var &  a )

inline

\[ \mbox{inv}(x) = \begin{cases} \frac{1}{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv}(x)}{\partial x} = \begin{cases} -\frac{1}{x^2} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 38 of file inv.hpp.

template<typename T_y , typename T_dof >

return_type_t<T_y, T_dof> stan::math::inv_chi_square_lpdf ( const T_y &  y, const T_dof &  nu  )

inline

Definition at line 123 of file inv_chi_square_lpdf.hpp.

template<typename T >

fvar<T> stan::math::inv_cloglog ( const fvar< T > &  x )

inline

Definition at line 13 of file inv_cloglog.hpp.

template<typename T >

apply_scalar_unary<inv_cloglog_fun, T>::return_t stan::math::inv_cloglog ( const T &  x )

inline

Vectorized version of inv_cloglog().

Template Parameters

T	type of container

Parameters

x

container

Returns

1 - exp(-exp()) applied to each value in x.

Definition at line 32 of file inv_cloglog.hpp.

var stan::math::inv_cloglog ( const var &  a )

inline

Return the inverse complementary log-log function applied specified variable (stan).

See inv_cloglog() for the double-based version.

The derivative is given by

\(\frac{d}{dx} \mbox{cloglog}^{-1}(x) = \exp (x - \exp (x))\).

Parameters

a	Variable argument.

Returns

The inverse complementary log-log of the specified argument.

Definition at line 37 of file inv_cloglog.hpp.

double stan::math::inv_cloglog ( double  x )

inline

The inverse complementary log-log function.

The function is defined by

inv_cloglog(x) = 1 - exp(-exp(x)).

This function can be used to implement the inverse link function for complementary-log-log regression.

\[ \mbox{inv\_cloglog}(y) = \begin{cases} \mbox{cloglog}^{-1}(y) & \mbox{if } -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_cloglog}(y)}{\partial y} = \begin{cases} \frac{\partial\, \mbox{cloglog}^{-1}(y)}{\partial y} & \mbox{if } -\infty\leq y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \mbox{cloglog}^{-1}(y) = 1 - \exp \left( - \exp(y) \right) \]

\[ \frac{\partial \, \mbox{cloglog}^{-1}(y)}{\partial y} = \exp(y-\exp(y)) \]

Parameters

x

Argument.

Returns

Inverse complementary log-log of the argument.

Definition at line 48 of file inv_cloglog.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::inv_gamma_lccdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	beta
	)

Definition at line 19 of file inv_gamma_lccdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::inv_gamma_lcdf	(	const T_y &	y,
		const T_shape &	alpha,
		const T_scale &	beta
	)

Definition at line 19 of file inv_gamma_lcdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::inv_gamma_lpdf ( const T_y &  y, const T_shape &  alpha, const T_scale &  beta  )

inline

Definition at line 142 of file inv_gamma_lpdf.hpp.

template<typename T >

fvar<T> stan::math::inv_logit ( const fvar< T > &  x )

inline

Returns the inverse logit function applied to the argument.

Template Parameters

T	scalar type of forward-mode autodiff variable argument.

Parameters

x

argument

Returns

inverse logit of argument

Definition at line 21 of file inv_logit.hpp.

template<typename T >

apply_scalar_unary<inv_logit_fun, T>::return_t stan::math::inv_logit ( const T &  x )

inline

Vectorized version of inv_logit().

Template Parameters

T	type of container

Parameters

x

container

Returns

Inverse logit applied to each value in x.

Definition at line 32 of file inv_logit.hpp.

var stan::math::inv_logit ( const var &  a )

inline

The inverse logit function for variables (stan).

See inv_logit() for the double-based version.

The derivative of inverse logit is

\(\frac{d}{dx} \mbox{logit}^{-1}(x) = \mbox{logit}^{-1}(x) (1 - \mbox{logit}^{-1}(x))\).

Parameters

a	Argument variable.

Returns

Inverse logit of argument.

Definition at line 32 of file inv_logit.hpp.

double stan::math::inv_logit ( double  a )

inline

Returns the inverse logit function applied to the argument.

The inverse logit function is defined by

\(\mbox{logit}^{-1}(x) = \frac{1}{1 + \exp(-x)}\).

This function can be used to implement the inverse link function for logistic regression.

The inverse to this function is logit.

\[ \mbox{inv\_logit}(y) = \begin{cases} \mbox{logit}^{-1}(y) & \mbox{if } -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_logit}(y)}{\partial y} = \begin{cases} \frac{\partial\, \mbox{logit}^{-1}(y)}{\partial y} & \mbox{if } -\infty\leq y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \mbox{logit}^{-1}(y) = \frac{1}{1 + \exp(-y)} \]

\[ \frac{\partial \, \mbox{logit}^{-1}(y)}{\partial y} = \frac{\exp(y)}{(\exp(y)+1)^2} \]

Parameters

a

Argument.

Returns

Inverse logit of argument.

Definition at line 49 of file inv_logit.hpp.

template<typename T >

fvar<T> stan::math::inv_Phi ( const fvar< T > &  p )

inline

Definition at line 15 of file inv_Phi.hpp.

double stan::math::inv_Phi ( double  p )

inline

The inverse of the unit normal cumulative distribution function.

The return value for a specified input probability, $p$, is the unit normal variate, $x$, such that

\(\Phi(x) = \int_{-\infty}^x \mbox{\sf Norm}(x|0, 1) \ dx = p\)

Algorithm first derived in 2003 by Peter Jon Aklam at http://home.online.no/~pjacklam/notes/invnorm/

Parameters

p	Argument between 0 and 1.

Returns

Real number

Definition at line 29 of file inv_Phi.hpp.

template<typename T >

apply_scalar_unary<inv_Phi_fun, T>::return_t stan::math::inv_Phi ( const T &  x )

inline

Vectorized version of inv_Phi().

Template Parameters

T	type of container

Parameters

x	variables in range [0, 1]

Returns

Inverse unit normal CDF of each value in x.

Exceptions

std::domain_error

if any value is not between 0 and 1.

Definition at line 34 of file inv_Phi.hpp.

var stan::math::inv_Phi ( const var &  p )

inline

The inverse of unit normal cumulative density function.

See inv_Phi() for the double-based version.

The derivative is the reciprocal of unit normal density function,

Parameters

p	Probability

Returns

The unit normal inverse cdf evaluated at p

Definition at line 34 of file inv_Phi.hpp.

double stan::math::inv_sqrt ( double  x )

inline

Definition at line 11 of file inv_sqrt.hpp.

template<typename T >

fvar<T> stan::math::inv_sqrt ( const fvar< T > &  x )

inline

Definition at line 12 of file inv_sqrt.hpp.

template<typename T >

apply_scalar_unary<inv_sqrt_fun, T>::return_t stan::math::inv_sqrt ( const T &  x )

inline

Vectorized version of inv_sqrt().

Template Parameters

T	type of container

Parameters

x

container

Returns

1 / sqrt of each value in x.

Definition at line 32 of file inv_sqrt.hpp.

var stan::math::inv_sqrt ( const var &  a )

inline

\[ \mbox{inv\_sqrt}(x) = \begin{cases} \frac{1}{\sqrt{x}} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_sqrt}(x)}{\partial x} = \begin{cases} -\frac{1}{2\sqrt{x^3}} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 41 of file inv_sqrt.hpp.

double stan::math::inv_square ( double  x )

inline

Definition at line 10 of file inv_square.hpp.

template<typename T >

fvar<T> stan::math::inv_square ( const fvar< T > &  x )

inline

Definition at line 12 of file inv_square.hpp.

template<typename T >

apply_scalar_unary<inv_square_fun, T>::return_t stan::math::inv_square ( const T &  x )

inline

Vectorized version of inv_square().

Template Parameters

T	type of container

Parameters

x

container

Returns

1 / the square of each value in x.

Definition at line 32 of file inv_square.hpp.

var stan::math::inv_square ( const var &  a )

inline

\[ \mbox{inv\_square}(x) = \begin{cases} \frac{1}{x^2} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_square}(x)}{\partial x} = \begin{cases} -\frac{2}{x^3} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 40 of file inv_square.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::inv_wishart_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  W, const T_dof &  nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &  S  )

inline

Definition at line 100 of file inv_wishart_lpdf.hpp.

template<class RNG >

Eigen::MatrixXd stan::math::inv_wishart_rng ( double  nu, const Eigen::MatrixXd &  S, RNG &  rng  )

inline

Definition at line 13 of file inv_wishart_rng.hpp.

template<typename T >

void stan::math::invalid_argument ( const char *  function, const char *  name, const T &  y, const char *  msg1, const char *  msg2  )

inline

Throw an invalid_argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid argument. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name> <msg1><y><msg2>"

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
msg1	Message to print before the variable
msg2	Message to print after the variable

Exceptions

std::invalid_argument

Definition at line 27 of file invalid_argument.hpp.

template<typename T >

void stan::math::invalid_argument ( const char *  function, const char *  name, const T &  y, const char *  msg1  )

inline

Throw an invalid_argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid argument. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.) The message is: "<function>: <name> <msg1><y>"

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
msg1	Message to print before the variable

Exceptions

std::invalid_argument

Definition at line 49 of file invalid_argument.hpp.

template<typename T >

void stan::math::invalid_argument_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg1, const char *  msg2  )

inline

Throw an invalid argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid arguments. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.) The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Index
msg1	Message to print before the variable
msg2	Message to print after the variable

Exceptions

std::invalid_argument

Definition at line 32 of file invalid_argument_vec.hpp.

template<typename T >

void stan::math::invalid_argument_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg  )

inline

Throw an invalid argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid arguments. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.) The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Index
msg	Message to print before the variable

Exceptions

std::invalid_argument

Definition at line 60 of file invalid_argument_vec.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::inverse ( const Eigen::Matrix< fvar< T >, R, C > &  m )

inline

Definition at line 16 of file inverse.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::inverse ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the inverse of the specified matrix.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	Specified matrix.

Returns

Inverse of the matrix.

Definition at line 21 of file inverse.hpp.

matrix_v stan::math::inverse ( const matrix_v &  m )

inline

Reverse mode specialization of calculating the inverse of the matrix.

Parameters

m	Specified matrix.

Returns

Inverse of the matrix.

Definition at line 76 of file inverse.hpp.

template<typename Vector >

void stan::math::inverse_softmax	(	const Vector &	simplex,
		Vector &	y
	)

Writes the inverse softmax of the simplex argument into the second argument.

See softmax for the inverse function and a definition of the relation.

The inverse softmax function is defined by

\(\mbox{inverse\_softmax}(x)[i] = \log x[i]\).

This function defines the inverse of softmax up to a scaling factor.

Because of the definition, values of 0.0 in the simplex are converted to negative infinity, and values of 1.0 are converted to 0.0.

There is no check that the input vector is a valid simplex vector.

Parameters

simplex	Simplex vector input.
y	Vector into which the inverse softmax is written.

Exceptions

std::invalid_argument

if size of the input and output vectors differ.

Definition at line 35 of file inverse_softmax.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::inverse_spd ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m )

inline

Returns the inverse of the specified symmetric, pos/neg-definite matrix.

Template Parameters

T	type of elements in the matrix

Parameters

m	Specified matrix.

Returns

Inverse of the matrix.

Definition at line 18 of file inverse_spd.hpp.

template<typename T >

bool stan::math::is_aligned	(	T *	ptr,
		unsigned int	bytes_aligned
	)

Return true if the specified pointer is aligned on the number of bytes.

This doesn't really make sense other than for powers of 2.

Parameters

ptr	Pointer to test.
bytes_aligned	Number of bytes of alignment required.

Returns

true if pointer is aligned.

Template Parameters

Type	of object to which pointer points.

Definition at line 29 of file stack_alloc.hpp.

template<typename T >

bool stan::math::is_any_nan ( T &&  x )

inline

Returns true if the input is NaN and false otherwise.

Delegates to stan::math::is_nan so that appropriate specializations can be loaded for autodiff types.

Parameters

x	Value to test.

Returns

true if the value is NaN.

Definition at line 21 of file is_any_nan.hpp.

template<typename T , typename... Ts>

bool stan::math::is_any_nan ( T &&  x, Ts &&...  xs  )

inline

Returns true if any input is NaN and false otherwise.

Delegates to stan::math::is_nan.

Parameters

x	first argument
xs	parameter pack of remaining arguments to forward to function

Returns

true if any value is NaN

Definition at line 35 of file is_any_nan.hpp.

template<typename T_y >

bool stan::math::is_cholesky_factor ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y )

inline

Return true if y is a valid Choleksy factor, if number of rows is not less than the number of columns, if there are no 0 columns, and no element in matrix is NaN.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. Note that Cholesky factors need not be square, but requires at least as many rows M as columns N (i.e., M >= N).

Template Parameters

T_y	Type of elements of Cholesky factor, requires class method .rows(), .cols(), and .diagonal()

Parameters

y	Matrix to test

Returns

true if y is a valid Choleksy factor, if number of rows is not less than the number of columns, if there are no 0 columns, and no element in matrix is NaN

Definition at line 29 of file is_cholesky_factor.hpp.

template<typename T_y >

bool stan::math::is_cholesky_factor_corr ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y )

inline

Return true if y is a valid Cholesky factor, if the number of rows is not less than the number of columns, if there are no zero columns, and no element in matrix is NaN.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. This definition does not require a square matrix.

Template Parameters

T_y	Type of elements of Cholesky factor, requires class methods .rows(), .row(), and .transpose()

Parameters

y	Matrix to test

Returns

true if y is a valid Cholesky factor, if the number of rows is not less than the number of columns, if there are no 0 columns, and no element in matrix is NaN

Definition at line 26 of file is_cholesky_factor_corr.hpp.

template<typename T_y , int R, int C>

bool stan::math::is_column_index ( const Eigen::Matrix< T_y, R, C > &  y, size_t  i  )

inline

Return true if column index is in bounds.

By default this is a 1-indexed check (as opposed to zero-indexed). Behavior can be changed by setting stan::error_index::value.

Template Parameters

T_y	Type of scalar, requires class method .cols()
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

y	matrix to test
i	column index to check

Returns

true if column index is in bounds

Definition at line 22 of file is_column_index.hpp.

template<typename T_y >

bool stan::math::is_corr_matrix ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y )

inline

Return true if the matrix is square and not 0x0, if the matrix is symmetric, diagonals are near 1, positive definite, and no elements are NaN A valid correlation matrix is symmetric, has a unit diagonal (all 1 values), and has all values between -1 and 1 (inclusive).

Template Parameters

T_y	Type of scalar, requires class method .rows() and .cols()

Parameters

y	Matrix to test

Returns

true if the matrix is square and not 0x0, if the matrix is symmetric, diagonals are near 1, positive definite, and no elements are NaN

Definition at line 30 of file is_corr_matrix.hpp.

bool stan::math::is_inf ( double  x )

inline

Returns true if the input is infinite and false otherwise.

Delegates to std::isinf.

Parameters

x	Value to test.

Returns

true if the value is infinite.

Definition at line 18 of file is_inf.hpp.

template<typename T >

int stan::math::is_inf ( const fvar< T > &  x )

inline

Returns 1 if the input's value is infinite and 0 otherwise.

Delegates to is_inf.

Parameters

x	Value to test.

Returns

1 if the value is infinite and 0 otherwise.

Definition at line 20 of file is_inf.hpp.

int stan::math::is_inf ( const var &  v )

inline

Returns 1 if the input's value is infinite and 0 otherwise.

Delegates to is_inf.

Parameters

v	Value to test.

Returns

1 if the value is infinite and 0 otherwise.

Definition at line 20 of file is_inf.hpp.

template<typename T >

bool stan::math::is_integer ( T  x )

inline

Returns true if the input is an integer and false otherwise.

Parameters

x	Value to test.

Returns

true if the value is an integer

Definition at line 17 of file is_integer.hpp.

template<typename T , int R, int C>

bool stan::math::is_ldlt_factor ( LDLT_factor< T, R, C > &  A )

inline

Return true if the specified LDLT factor is invalid.

An LDLT_factor is invalid if it was constructed from a matrix that is not positive definite. The check is that the .success() method returns true.

Template Parameters

T	Type of scalar, requires class access to .success()
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

A	The LDLT factor to check for validity

Returns

true if the LDLT factor is valid

Definition at line 23 of file is_ldlt_factor.hpp.

template<typename T_y , typename T_high >

bool stan::math::is_less_or_equal ( const T_y &  y, const T_high &  high  )

inline

Return true if y is less or equal to high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters

T_y	Type of y
T_high	Type of upper bound

Parameters

y	Variable to check
high	Upper bound

Returns

true if y is less than or equal to low and if and element of y or high is NaN

Definition at line 22 of file is_less_or_equal.hpp.

template<typename T_y >

bool stan::math::is_lower_triangular ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y )

inline

Return true is matrix is lower triangular.

A matrix x is not lower triangular if there is a non-zero entry x[m, n] with m < n. This function only inspect the upper and triangular portion of the matrix, not including the diagonal.

Template Parameters

T	Type of scalar of the matrix

Parameters

y	Matrix to test

Returns

true is matrix is lower triangular

Definition at line 24 of file is_lower_triangular.hpp.

template<typename T , int R, int C>

bool stan::math::is_mat_finite ( const Eigen::Matrix< T, R, C > &  y )

inline

Return true is the specified matrix is finite.

Template Parameters

T	Scalar type of the matrix, requires class method .allFinite()
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

y	Matrix to test

Returns

true if the matrix is finite

Definition at line 20 of file is_mat_finite.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

bool stan::math::is_matching_dims ( const Eigen::Matrix< T1, R1, C1 > &  y1, const Eigen::Matrix< T2, R2, C2 > &  y2  )

inline

Return true if the two matrices are of the same size.

This function checks the runtime sizes only.

Template Parameters

T1	Scalar type of the first matrix, requires class method .size()
T2	Scalar type of the second matrix, requires class method .size()
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

y1	first matrix to test
y2	second matrix to test

Returns

true if the dimensions of the matrices match

Definition at line 27 of file is_matching_dims.hpp.

template<bool check_compile, typename T1 , typename T2 , int R1, int C1, int R2, int C2>

bool stan::math::is_matching_dims ( const Eigen::Matrix< T1, R1, C1 > &  y1, const Eigen::Matrix< T2, R2, C2 > &  y2  )

inline

Return true if the two matrices are of the same size.

This function checks the runtime sizes and can also check the static sizes as well. For example, a 4x1 matrix is not the same as a vector with 4 elements.

Template Parameters

check_compile	Whether to check the static sizes
T1	Scalar type of the first matrix
T2	Scalar type of the second matrix
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

y1	first matrix to test
y2	second matrix to test

Returns

true if the dimensions of the matrices match

Definition at line 51 of file is_matching_dims.hpp.

template<typename T_y1 , typename T_y2 >

bool stan::math::is_matching_size ( const T_y1 &  y1, const T_y2 &  y2  )

inline

Return true if two structures are the same size.

This is a top-level sizing function for std::vector where a 2x3 vector returns 2,and a total sizing function for Eigen::Matrix where a 2x3 matrix will return 6.

Template Parameters

T_y1	Type of the first variable, requires class method .size()
T_y2	Type of the second variable, requires class method .size()

Parameters

y1	First variable
y2	Second variable

Returns

true if the variable sizes match

Definition at line 24 of file is_matching_size.hpp.

template<typename T , typename = require_arithmetic_t<T>>

bool stan::math::is_nan ( T  x )

inline

Returns true if the input is NaN and false otherwise.

Delegates to std::isnan.

Parameters

x	Value to test.

Returns

true if the value is NaN.

Definition at line 19 of file is_nan.hpp.

template<typename T , require_fvar_t< T > ...>

bool stan::math::is_nan ( T &&  v )

inline

Returns 1 if the input's value is NaN and 0 otherwise.

Delegates to is_nan.

Parameters

x	Value to test.

Returns

1 if the value is NaN and 0 otherwise.

Delegates to is_nan(double).

Parameters

v	Value to test.

Returns

1 if the value is NaN and 0 otherwise.

Definition at line 21 of file is_nan.hpp.

template<typename T >

bool stan::math::is_nonpositive_integer ( T  x )

inline

Returns true if the input is a nonpositive integer and false otherwise.

Parameters

x	Value to test.

Returns

true if the value is an integer

Definition at line 17 of file is_nonpositive_integer.hpp.

template<typename T_y >

bool stan::math::is_nonzero_size ( const T_y &  y )

inline

Returns true if the specified matrix/vector is size nonzero.

Template Parameters

T_y	Type of container, requires class method .size()

Parameters

y	Container to test – matrix/vector

Returns

true if container has size zero

Definition at line 15 of file is_nonzero_size.hpp.

template<typename T_y >

bool stan::math::is_not_nan ( const T_y &  y )

inline

Return true if y is not NaN.

This function is vectorized and will check each element of y. If no element is NaN, this function will return true.

Template Parameters

T_y

Type of y

Parameters

y	Variable to check

Returns

true if no element of y is NaN

Definition at line 21 of file is_not_nan.hpp.

template<typename T_y >

bool stan::math::is_ordered ( const std::vector< T_y > &  y )

inline

Return true if the vector is sorted into strictly increasing order.

Template Parameters

T_y	Type of scalar, requires class method .size()

Parameters

y	std::vector to test

Returns

true if vector is sorted in ascending order

Definition at line 19 of file is_ordered.hpp.

template<typename T_y >

bool stan::math::is_pos_definite ( const Eigen::Matrix< T_y,-1,-1 > &  y )

inline

Return true if the matrix is square or if the matrix has non-zero size, or if the matrix is symmetric, or if it is positive definite, or if no element is NaN.

Template Parameters

T_y	Type of scalar of the matrix, requires class method .rows()

Parameters

y	Matrix to test

Returns

true if the matrix is square or if the matrix has non-0 size, or if the matrix is symmetric, or if it is positive definite, or if no element is NaN

Definition at line 27 of file is_pos_definite.hpp.

template<typename Derived >

bool stan::math::is_pos_definite ( const Eigen::LDLT< Derived > &  cholesky )

inline

Return true if the matrix is positive definite.

Template Parameters

Derived

Derived type of the Eigen::LDLT transform, requires class method .info() and .isPositive() and .vectorD().array()

Parameters

cholesky

Eigen::LDLT to test, whose progenitor must not have any NaN elements

Returns

true if the matrix is positive definite

Definition at line 55 of file is_pos_definite.hpp.

template<typename Derived >

bool stan::math::is_pos_definite ( const Eigen::LLT< Derived > &  cholesky )

inline

Return true if diagonal of the L matrix is positive.

Template Parameters

Derived

Derived type of the Eigen::LLT transform, requires class method .info() and .matrixLLT().diagonal().array()

Parameters

cholesky

Eigen::LLT to test, whose progenitor must not have any NaN elements

Returns

true if diagonal of the L matrix is positive

Definition at line 70 of file is_pos_definite.hpp.

template<typename T_y >

bool stan::math::is_positive ( const T_y &  y )

inline

Return true if y is positive.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

y	Variable to check

Returns

true if vector contains only positive elements

Definition at line 18 of file is_positive.hpp.

bool stan::math::is_positive ( int  size )

inline

Return true if size is positive.

Parameters

size	Size value to check

Returns

true if size is not zero or negative

Definition at line 32 of file is_positive.hpp.

template<typename T_y >

bool stan::math::is_scal_finite ( const T_y &  y )

inline

Return true if y is finite.

This function is vectorized and will check each element of y.

Template Parameters

T_y

Type of y

Parameters

y	Variable to check

Exceptions

<code>true</code>

if y is not infinity, -infinity, or NaN

Definition at line 20 of file is_scal_finite.hpp.

template<typename T_size1 , typename T_size2 >

bool stan::math::is_size_match ( T_size1  i, T_size2  j  )

inline

Return true if the provided sizes match.

Template Parameters

T_size1	Type of size 1
T_size2	Type of size 2

Parameters

i	Size of variable 1
j	Size of variable 2

Returns

true if provided dimensions match

Definition at line 17 of file is_size_match.hpp.

template<typename T_y >

bool stan::math::is_square ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y )

inline

Return true if the matrix is square.

This check allows 0x0 matrices.

Template Parameters

T	Type of scalar, requires class method .rows() and .cols()

Parameters

y	Matrix to test

Returns

true if matrix is square

Definition at line 20 of file is_square.hpp.

template<typename T_y >

bool stan::math::is_symmetric ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y )

inline

Return true if the matrix is square, and no element not on the main diagonal is NaN.

Template Parameters

T_y	Type of scalar, requires class method .rows()

Parameters

y	Matrix to test

Returns

true if the matrix is square, and no element not on the main diagonal is NaN

Definition at line 22 of file is_symmetric.hpp.

template<typename T >

bool stan::math::is_uninitialized ( T  x )

inline

Returns true if the specified variable is uninitialized.

Arithmetic types are always initialized by definition (the value is not specified).

Template Parameters

T	Type of object to test.

Parameters

x	Object to test.

Returns

true if the specified object is uninitialized.

false if input is NaN.

Definition at line 19 of file is_uninitialized.hpp.

bool stan::math::is_uninitialized ( var  x )

inline

Returns true if the specified variable is uninitialized.

This overload of the is_uninitialized() function delegates the return to the is_uninitialized() method on the specified variable.

Parameters

x	Object to test.

Returns

true if the specified object is uninitialized.

Definition at line 23 of file is_uninitialized.hpp.

template<typename T_prob >

bool stan::math::is_unit_vector ( const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta )

inline

Return true if the vector is not a unit vector or if any element is NaN.

A valid unit vector is one where the square elements summed is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accpets Eigen::Matrix vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters

T_prob

Scalar type of the vector, reqires class method .squaredNorm()

Parameters

theta

Eigen vector to test

Returns

true if the vector is not a unit vector or if any element is NaN

Definition at line 27 of file is_unit_vector.hpp.

template<typename T , typename F >

void stan::math::jacobian	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	J
	)

Definition at line 11 of file jacobian.hpp.

template<typename F >

void stan::math::jacobian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	J
	)

Definition at line 14 of file jacobian.hpp.

template<typename F1 , typename F2 = kinsol_J_f>

Eigen::VectorXd stan::math::kinsol_solve	(	const F1 &	f,
		const Eigen::VectorXd &	x,
		const Eigen::VectorXd &	y,
		const std::vector< double > &	dat,
		const std::vector< int > &	dat_int,
		std::ostream *	msgs = nullptr,
		double	scaling_step_tol = 1e-3,
		double	function_tolerance = 1e-6,
		long int	max_num_steps = 200,
		bool	custom_jacobian = 1,
		const F2 &	J_f = kinsol_J_f(),
		int	steps_eval_jacobian = 10,
		int	global_line_search = KIN_LINESEARCH
	)

Return the solution to the specified algebraic system, given an initial guess.

Invokes the Kinsol solver from Sundials.

Template Parameters

F	type of equation system function.
T	type of initial guess vector.

Parameters

[in]	f	Functor that evaluated the system of equations.
[in]	x	Vector of starting values.
[in]	y	Parameter vector for the equation system. The function is overloaded to treat y as a vector of doubles or of a a template type T.
[in]	dat	Continuous data vector for the equation system.
[in]	dat_int	Integer data vector for the equation system.
[in,out]	msgs	The print stream for warning messages.
[in]	scaling_step_tol	Scaled-step stopping tolerance. If a Newton step is smaller than the scaling step tolerance, the code breaks, assuming the solver is no longer making significant progress (i.e. is stuck)
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	Maximum number of function evaluations.
[in]	custom_jacobian	If 0, use Kinsol's default to compute the jacobian for the Newton step, namely Quotient Difference (finite difference). If 1, use reverse-mode AD, unless the user specifies their own method.
[in]	J_f	A functor that computes the Jacobian for the Newton step. Defaults to reverse-mode AD.
[in]	steps_eval_jacobian	Maximum number of steps before the Jacobian gets recomputed. Note that Kinsol's default is 10. If equal to 1, the algorithm computes exact Newton steps.
[in]	global_line_search	does the solver use a global line search? If equal to KIN_NONE, no, if KIN_LINESEARCH, yes.

Returns

x_solution Vector of solutions to the system of equations.

Exceptions

<code>std::invalid_argument</code>	if Kinsol returns a negative flag when setting up the solver.
<code>boost::math::evaluation_error</code>	if Kinsol returns a negative flag after attempting to solve the equation.

Definition at line 58 of file kinsol_solve.hpp.

template<typename T , typename L >

return_type_t<T, L> stan::math::lb_constrain ( const T &  x, const L &  lb  )

inline

Return the lower-bounded value for the specified unconstrained input and specified lower bound.

The transform applied is

\(f(x) = \exp(x) + L\)

where \(L\) is the constant lower bound.

If the lower bound is negative infinity, this function reduces to identity_constrain(x).

Template Parameters

T	type of scalar
L	type of lower bound

Parameters

[in]	x	Unconstrained scalar input
[in]	lb	lower bound on constrained ouptut

Returns

lower bound constrained value correspdonding to inputs

Definition at line 32 of file lb_constrain.hpp.

template<typename T , typename L >

return_type_t<T, L> stan::math::lb_constrain ( const T &  x, const L &  lb, T &  lp  )

inline

Return the lower-bounded value for the speicifed unconstrained input and specified lower bound, incrementing the specified reference with the log absolute Jacobian determinant of the transform.

If the lower bound is negative infinity, this function reduces to identity_constraint(x, lp).

Template Parameters

T	type of scalar.
L	type of lower bound.

Parameters

[in]	x	unconstrained scalar input
[in]	lb	lower bound on output
[in,out]	lp	Reference to log probability to increment.

Returns

lower-bound constrained value corresponding to inputs

Definition at line 57 of file lb_constrain.hpp.

template<typename T , typename L >

return_type_t<T, L> stan::math::lb_free ( const T &  y, const L &  lb  )

inline

Return the unconstrained value that produces the specified lower-bound constrained value.

If the lower bound is negative infinity, it is ignored and the function reduces to identity_free(y).

Template Parameters

T	type of scalar
L	type of lower bound

Parameters

[in]	y	input scalar
[in]	lb	lower bound

Returns

unconstrained value that produces the input when constrained

Exceptions

std::domain_error

if y is lower than the lower bound

Definition at line 29 of file lb_free.hpp.

template<typename T >

fvar<T> stan::math::lbeta ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file lbeta.hpp.

template<typename T >

fvar<T> stan::math::lbeta ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 21 of file lbeta.hpp.

template<typename T >

fvar<T> stan::math::lbeta ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 27 of file lbeta.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::lbeta ( const T1  a, const T2  b  )

inline

Return the log of the beta function applied to the specified arguments.

The beta function is defined for \(a > 0\) and \(b > 0\) by

\(\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\).

This function returns its log,

\(\log \mbox{B}(a, b) = \log \Gamma(a) + \log \Gamma(b) - \log \Gamma(a+b)\).

See stan::math::lgamma() for the double-based and stan::math for the variable-based log Gamma function.

\[ \mbox{lbeta}(\alpha, \beta) = \begin{cases} \ln\int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lbeta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \Psi(\alpha)-\Psi(\alpha+\beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lbeta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \Psi(\beta)-\Psi(\alpha+\beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Parameters

a	First value
b	Second value

Returns

Log of the beta function applied to the two values.

Template Parameters

T1	Type of first value.
T2	Type of second value.

Definition at line 57 of file lbeta.hpp.

var stan::math::lbeta ( const var &  a, const var &  b  )

inline

Definition at line 64 of file lbeta.hpp.

var stan::math::lbeta ( const var &  a, double  b  )

inline

Definition at line 83 of file lbeta.hpp.

var stan::math::lbeta ( double  a, const var &  b  )

inline

Definition at line 102 of file lbeta.hpp.

var stan::math::ldexp ( const var &  a, int  b  )

inline

Definition at line 19 of file ldexp.hpp.

template<typename T1 , typename = require_arithmetic_t<T1>>

double stan::math::ldexp ( T1  a, int  b  )

inline

Returns the product of a (the significand) and 2 to power b (the exponent).

Template Parameters

T	Scalar type of significand

Parameters

[in]	a	the significand
[in]	b	an integer that is the exponent

Returns

product of a times 2 to the power b

Definition at line 20 of file ldexp.hpp.

template<typename T >

fvar<T> stan::math::ldexp ( const fvar< T > &  a, int  b  )

inline

Returns the product of a (the significand) times 2 to power b (the exponent).

Template Parameters

T	Scalar type of significand

Parameters

[in]	a	the significand
[in]	b	an integer that is the exponent

Returns

product of a times 2 to the power b

Definition at line 21 of file ldexp.hpp.

template<typename T >

fvar<T> stan::math::lgamma ( const fvar< T > &  x )

inline

Return the natural logarithm of the gamma function applied to the specified argument.

Template Parameters

T	Scalar type of autodiff variable.

Parameters

x

Argument.

Returns

natural logarithm of the gamma function of argument.

Definition at line 21 of file lgamma.hpp.

var stan::math::lgamma ( const var &  a )

inline

The log gamma function for variables (C99).

The derivatie is the digamma function,

\(\frac{d}{dx} \Gamma(x) = \psi^{(0)}(x)\).

Parameters

a	The variable.

Returns

Log gamma of the variable.

Definition at line 30 of file lgamma.hpp.

template<typename T >

apply_scalar_unary<lgamma_fun, T>::return_t stan::math::lgamma ( const T &  x )

inline

Vectorized version of lgamma().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural log of the gamma function applied to each value in x.

Exceptions

std::domain_error

if any value is a negative integer or 0.

Definition at line 35 of file lgamma.hpp.

double stan::math::lgamma ( double  x )

inline

Return the natural logarithm of the gamma function applied to the specified argument.

\[ \mbox{lgamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \ln\Gamma(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lgamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

x

argument

Returns

natural logarithm of the gamma function applied to argument

Definition at line 62 of file lgamma.hpp.

double stan::math::lgamma ( int  x )

inline

Return the natural logarithm of the gamma function applied to the specified argument.

Parameters

x

argument

Returns

natural logarithm of the gamma function applied to argument

Definition at line 81 of file lgamma.hpp.

template<bool propto, typename T_covar , typename T_shape >

return_type_t<T_covar, T_shape> stan::math::lkj_corr_cholesky_lpdf	(	const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &	L,
		const T_shape &	eta
	)

Definition at line 16 of file lkj_corr_cholesky_lpdf.hpp.

template<typename T_covar , typename T_shape >

return_type_t<T_covar, T_shape> stan::math::lkj_corr_cholesky_lpdf ( const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  L, const T_shape &  eta  )

inline

Definition at line 54 of file lkj_corr_cholesky_lpdf.hpp.

template<class RNG >

Eigen::MatrixXd stan::math::lkj_corr_cholesky_rng ( size_t  K, double  eta, RNG &  rng  )

inline

Definition at line 13 of file lkj_corr_cholesky_rng.hpp.

template<bool propto, typename T_y , typename T_shape >

return_type_t<T_y, T_shape> stan::math::lkj_corr_lpdf	(	const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &	y,
		const T_shape &	eta
	)

Definition at line 46 of file lkj_corr_lpdf.hpp.

template<typename T_y , typename T_shape >

return_type_t<T_y, T_shape> stan::math::lkj_corr_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y, const T_shape &  eta  )

inline

Definition at line 79 of file lkj_corr_lpdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::lkj_cov_lpdf	(	const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &	y,
		const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &	mu,
		const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &	sigma,
		const T_shape &	eta
	)

Definition at line 16 of file lkj_cov_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::lkj_cov_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &  mu, const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &  sigma, const T_shape &  eta  )

inline

Definition at line 52 of file lkj_cov_lpdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::lkj_cov_lpdf	(	const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	eta
	)

Definition at line 64 of file lkj_cov_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::lkj_cov_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y, const T_loc &  mu, const T_scale &  sigma, const T_shape &  eta  )

inline

Definition at line 93 of file lkj_cov_lpdf.hpp.

template<typename T >

fvar<typename stan::return_type<T, int>::type> stan::math::lmgamma ( int  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file lmgamma.hpp.

var stan::math::lmgamma ( int  a, const var &  b  )

inline

Definition at line 27 of file lmgamma.hpp.

template<typename T >

return_type_t<T> stan::math::lmgamma ( int  k, T  x  )

inline

Return the natural logarithm of the multivariate gamma function with the speciifed dimensions and argument.

The multivariate gamma function \(\Gamma_k(x)\) for dimensionality \(k\) and argument \(x\) is defined by

\(\Gamma_k(x) = \pi^{k(k-1)/4} \, \prod_{j=1}^k \Gamma(x + (1 - j)/2)\),

where \(\Gamma()\) is the gamma function.

\[ \mbox{lmgamma}(n, x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \ln\Gamma_n(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lmgamma}(n, x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \ln\Gamma_n(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \ln\Gamma_n(x) = \pi^{n(n-1)/4} \, \prod_{j=1}^n \Gamma(x + (1 - j)/2) \]

\[ \frac{\partial \, \ln\Gamma_n(x)}{\partial x} = \sum_{j=1}^n \Psi(x + (1 - j) / 2) \]

Parameters

k	Number of dimensions.
x	Function argument.

Returns

Natural log of the multivariate gamma function.

Template Parameters

T	Type of scalar.

Definition at line 54 of file lmgamma.hpp.

template<typename T >

fvar<T> stan::math::log ( const fvar< T > &  x )

inline

Definition at line 13 of file log.hpp.

double stan::math::log ( int  x )

inline

Return the natural log of the specified argument.

This version is required to disambiguate log(int).

Parameters

[in]

x

Argument.

Returns

Natural log of argument.

Definition at line 17 of file log.hpp.

template<typename T >

apply_scalar_unary<log_fun, T>::return_t stan::math::log ( const T &  x )

inline

Return the elementwise natural log of the specified argument, which may be a scalar or any Stan container of numeric scalars.

The return type is the same as the argument type.

Template Parameters

T	type of container

Parameters

[in]

x

container

Returns

Elementwise application of natural log to the argument.

Definition at line 39 of file log.hpp.

var stan::math::log ( const var &  a )

inline

Return the natural log of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \log x = \frac{1}{x}\).

\[ \mbox{log}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \ln(x) & \mbox{if } x \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \frac{1}{x} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable whose log is taken.

Returns

Natural log of variable.

Definition at line 47 of file log.hpp.

template<typename T >

fvar<T> stan::math::log10 ( const fvar< T > &  x )

inline

Definition at line 13 of file log10.hpp.

template<typename T >

apply_scalar_unary<log10_fun, T>::return_t stan::math::log10 ( const T &  x )

inline

Vectorized version of log10().

Template Parameters

T	type of container

Parameters

x

container

Returns

Log base-10 applied to each value in x.

Definition at line 33 of file log10.hpp.

var stan::math::log10 ( const var &  a )

inline

Return the base 10 log of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \log_{10} x = \frac{1}{x \log 10}\).

\[ \mbox{log10}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \log_{10}(x) & \mbox{if } x \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log10}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \frac{1}{x \ln10} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable whose log is taken.

Returns

Base 10 log of variable.

Definition at line 51 of file log10.hpp.

double stan::math::log10 ( )

inline

Return natural logarithm of ten.

Returns

Natural logarithm of ten.

Definition at line 99 of file constants.hpp.

template<typename T >

fvar<T> stan::math::log1m ( const fvar< T > &  x )

inline

Definition at line 12 of file log1m.hpp.

var stan::math::log1m ( const var &  a )

inline

The log (1 - x) function for variables.

The derivative is given by

\(\frac{d}{dx} \log (1 - x) = -\frac{1}{1 - x}\).

Parameters

a	The variable.

Returns

The variable representing log of 1 minus the variable.

Definition at line 29 of file log1m.hpp.

template<typename T >

apply_scalar_unary<log1m_fun, T>::return_t stan::math::log1m ( const T &  x )

inline

Vectorized version of log1m().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural log of 1 minus each value in x.

Definition at line 32 of file log1m.hpp.

double stan::math::log1m ( double  x )

inline

Return the natural logarithm of one minus the specified value.

The main use of this function is to cut down on intermediate values during algorithmic differentiation.

\[ \mbox{log1m}(x) = \begin{cases} \ln(1-x) & \mbox{if } x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1m}(x)}{\partial x} = \begin{cases} -\frac{1}{1-x} & \mbox{if } x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

[in]

x

Argument.

Returns

Natural log of one minus the argument.

Exceptions

<code>std::domain_error</code>	If the argument is greater than 1.
<code>std::overflow_error</code>	If the computation overflows.

Definition at line 42 of file log1m.hpp.

template<typename T >

fvar<T> stan::math::log1m_exp ( const fvar< T > &  x )

inline

Return the natural logarithm of one minus the exponentiation of the specified argument.

Template Parameters

T	Scalar type of autodiff variable.

Parameters

x

Argument.

Returns

log of one minus the exponentiation of the argument.

Definition at line 23 of file log1m_exp.hpp.

template<typename T >

apply_scalar_unary<log1m_exp_fun, T>::return_t stan::math::log1m_exp ( const T &  x )

inline

Vectorized version of log1m_exp().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural log of (1 - exp()) applied to each value in x.

Definition at line 32 of file log1m_exp.hpp.

var stan::math::log1m_exp ( const var &  x )

inline

Return the log of 1 minus the exponential of the specified variable.

The deriative of log(1 - exp(x)) with respect to x is -1 / expm1(-x).

Parameters

[in]

x

Argument.

Returns

Natural logarithm of one minus the exponential of the argument.

Definition at line 32 of file log1m_exp.hpp.

double stan::math::log1m_exp ( double  a )

inline

Calculates the natural logarithm of one minus the exponential of the specified value without overflow,.

log1m_exp(x) = log(1-exp(x))

This function is only defined for x < 0

\[ \mbox{log1m\_exp}(x) = \begin{cases} \ln(1-\exp(x)) & \mbox{if } x < 0 \\ \textrm{NaN} & \mbox{if } x \geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asinh}(x)}{\partial x} = \begin{cases} -\frac{\exp(x)}{1-\exp(x)} & \mbox{if } x < 0 \\ \textrm{NaN} & \mbox{if } x \geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

[in]

a

Argument.

Returns

natural logarithm of one minus the exponential of the argument.

Definition at line 45 of file log1m_exp.hpp.

var stan::math::log1m_inv_logit ( const var &  u )

inline

Return the natural logarithm of one minus the inverse logit of the specified argument.

Parameters

u

argument

Returns

log of one minus the inverse logit of the argument

Definition at line 20 of file log1m_inv_logit.hpp.

template<typename T >

fvar<T> stan::math::log1m_inv_logit ( const fvar< T > &  x )

inline

Return the natural logarithm of one minus the inverse logit of the specified argument.

Template Parameters

T	scalar type of forward-mode autodiff variable argument.

Parameters

x

argument

Returns

log of one minus the inverse logit of the argument

Definition at line 22 of file log1m_inv_logit.hpp.

double stan::math::log1m_inv_logit ( double  u )

inline

Returns the natural logarithm of 1 minus the inverse logit of the specified argument.

\[ \mbox{log1m\_inv\_logit}(x) = \begin{cases} -\ln(\exp(x)+1) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1m\_inv\_logit}(x)}{\partial x} = \begin{cases} -\frac{\exp(x)}{\exp(x)+1} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

u

argument

Returns

log of one minus the inverse logit of the argument

Definition at line 35 of file log1m_inv_logit.hpp.

template<typename T >

apply_scalar_unary<log1m_inv_logit_fun, T>::return_t stan::math::log1m_inv_logit ( const T &  x )

inline

Return the elementwise application of log1m_inv_logit() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise log1m_inv_logit of members of container.

Definition at line 40 of file log1m_inv_logit.hpp.

double stan::math::log1m_inv_logit ( int  u )

inline

Return the natural logarithm of one minus the inverse logit of the specified argument.

Parameters

u

argument

Returns

log of one minus the inverse logit of the argument

Definition at line 50 of file log1m_inv_logit.hpp.

template<typename T >

fvar<T> stan::math::log1p ( const fvar< T > &  x )

inline

Definition at line 12 of file log1p.hpp.

double stan::math::log1p ( double  x )

inline

Return the natural logarithm of one plus the specified value.

\[ \mbox{log1p}(x) = \log(1 + x) \]

This version is more stable for arguments near zero than the direct definition. If log1p(x) is defined to be negative infinity.

Parameters

[in]

x

Argument.

Returns

Natural log of one plus the argument.

Exceptions

std::domain_error

If argument is less than -1.

Definition at line 27 of file log1p.hpp.

var stan::math::log1p ( const var &  a )

inline

The log (1 + x) function for variables (C99).

The derivative is given by

\(\frac{d}{dx} \log (1 + x) = \frac{1}{1 + x}\).

Parameters

a	The variable.

Returns

The log of 1 plus the variable.

Definition at line 29 of file log1p.hpp.

template<typename T >

apply_scalar_unary<log1p_fun, T>::return_t stan::math::log1p ( const T &  x )

inline

Return the elementwise application of log1p() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

Elementwise log1p of members of container.

Definition at line 38 of file log1p.hpp.

double stan::math::log1p ( int  x )

inline

Return the natural logarithm of one plus the specified argument.

This version is required to disambiguate log1p(int).

Parameters

[in]

x

Argument.

Returns

Natural logarithm of one plus the argument.

Exceptions

std::domain_error

If argument is less than -1.

Definition at line 45 of file log1p.hpp.

template<typename T >

fvar<T> stan::math::log1p_exp ( const fvar< T > &  x )

inline

Definition at line 13 of file log1p_exp.hpp.

var stan::math::log1p_exp ( const var &  a )

inline

Return the log of 1 plus the exponential of the specified variable.

Definition at line 24 of file log1p_exp.hpp.

template<typename T >

apply_scalar_unary<log1p_exp_fun, T>::return_t stan::math::log1p_exp ( const T &  x )

inline

Vectorized version of log1m_exp().

Template Parameters

T	type of container

Parameters

x

container

Returns

Natural log of (1 + exp()) applied to each value in x.

Definition at line 32 of file log1p_exp.hpp.

double stan::math::log1p_exp ( double  a )

inline

Calculates the log of 1 plus the exponential of the specified value without overflow.

This function is related to other special functions by:

log1p_exp(x)

= log1p(exp(a))

= log(1 + exp(x))

= log_sum_exp(0, x).

\[ \mbox{log1p\_exp}(x) = \begin{cases} \ln(1+\exp(x)) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1p\_exp}(x)}{\partial x} = \begin{cases} \frac{\exp(x)}{1+\exp(x)} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 43 of file log1p_exp.hpp.

template<typename T >

fvar<T> stan::math::log2 ( const fvar< T > &  x )

inline

Return the base two logarithm of the specified argument.

Template Parameters

T	scalar type

Parameters

x

argument

Returns

base two logarithm of argument

Definition at line 20 of file log2.hpp.

template<typename T , typename = require_arithmetic_t<T>>

double stan::math::log2 ( T  u )

inline

Returns the base two logarithm of the argument (C99, C++11).

The function is defined by:

log2(a) = log(a) / std::log(2.0).

Parameters

[in]

u

argument

Returns

base two logarithm of argument

Definition at line 22 of file log2.hpp.

double stan::math::log2 ( )

inline

Return natural logarithm of two.

Returns

Natural logarithm of two.

Definition at line 32 of file log2.hpp.

template<typename T , typename = require_vector_like_t<T>>

apply_scalar_unary<log2_fun, T>::return_t stan::math::log2 ( const T &  x )

inline

Return the elementwise application of log2() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise log2 of container elements

Definition at line 39 of file log2.hpp.

var stan::math::log2 ( const var &  a )

inline

Returns the base 2 logarithm of the specified variable (C99).

See log2() for the double-based version.

The derivative is

\(\frac{d}{dx} \log_2 x = \frac{1}{x \log 2}\).

\[ \mbox{log2}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \\ \log_2(x) & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log2}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \\ \frac{1}{x\ln2} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Specified variable.

Returns

Base 2 logarithm of the variable.

Definition at line 50 of file log2.hpp.

template<int R, int C>

var stan::math::log_determinant ( const Eigen::Matrix< var, R, C > &  m )

inline

Definition at line 14 of file log_determinant.hpp.

template<typename T , int R, int C>

fvar<T> stan::math::log_determinant ( const Eigen::Matrix< fvar< T >, R, C > &  m )

inline

Definition at line 15 of file log_determinant.hpp.

template<typename T , int R, int C>

T stan::math::log_determinant ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the log absolute determinant of the specified square matrix.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	Specified matrix.

Returns

log absolute determinant of the matrix.

Exceptions

std::domain_error

if matrix is not square.

Definition at line 22 of file log_determinant.hpp.

template<int R, int C, typename T >

T stan::math::log_determinant_ldlt ( LDLT_factor< T, R, C > &  A )

inline

Returns log(abs(det(A))) given a LDLT_factor of A.

Template Parameters

T	type of elements in the LDLT_factor
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

A	LDLT_factor

Returns

the log(abs(det(A))

Definition at line 20 of file log_determinant_ldlt.hpp.

template<int R, int C>

var stan::math::log_determinant_ldlt

(

LDLT_factor< var, R, C > &

A

)

Definition at line 42 of file log_determinant_ldlt.hpp.

template<int R, int C>

var stan::math::log_determinant_spd ( const Eigen::Matrix< var, R, C > &  m )

inline

Definition at line 14 of file log_determinant_spd.hpp.

template<typename T , int R, int C>

T stan::math::log_determinant_spd ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the log absolute determinant of the specified square matrix.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	specified matrix

Returns

log absolute determinant of the matrix

Exceptions

std::domain_error

if matrix is not square and symmetric

Definition at line 23 of file log_determinant_spd.hpp.

template<typename T >

fvar<T> stan::math::log_diff_exp ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file log_diff_exp.hpp.

template<typename T1 , typename T2 >

fvar<T2> stan::math::log_diff_exp ( const T1 &  x1, const fvar< T2 > &  x2  )

inline

Definition at line 27 of file log_diff_exp.hpp.

template<typename T1 , typename T2 >

fvar<T1> stan::math::log_diff_exp ( const fvar< T1 > &  x1, const T2 &  x2  )

inline

Definition at line 38 of file log_diff_exp.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::log_diff_exp ( const T1  x, const T2  y  )

inline

The natural logarithm of the difference of the natural exponentiation of x and the natural exponentiation of y.

This function is only defined for x >= y

\[ \mbox{log\_diff\_exp}(x, y) = \begin{cases} \textrm{NaN} & \mbox{if } x < y\\ \ln(\exp(x)-\exp(y)) & \mbox{if } x \geq y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_diff\_exp}(x, y)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x \leq y\\ \frac{\exp(x)}{\exp(x)-\exp(y)} & \mbox{if } x > y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_diff\_exp}(x, y)}{\partial y} = \begin{cases} \textrm{NaN} & \mbox{if } x \leq y\\ -\frac{\exp(y)}{\exp(x)-\exp(y)} & \mbox{if } x > y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Definition at line 47 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( const var &  a, const var &  b  )

inline

Returns the log difference of the exponentiated arguments.

Parameters

[in]	a	First argument.
[in]	b	Second argument.

Returns

Log difference of the exponentiated arguments.

Definition at line 57 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( const var &  a, double  b  )

inline

Returns the log difference of the exponentiated arguments.

Parameters

[in]	a	First argument.
[in]	b	Second argument.

Returns

Log difference of the exponentiated arguments.

Definition at line 68 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( double  a, const var &  b  )

inline

Returns the log difference of the exponentiated arguments.

Parameters

[in]	a	First argument.
[in]	b	Second argument.

Returns

Log difference of the exponentiated arguments.

Definition at line 79 of file log_diff_exp.hpp.

template<typename T >

fvar<T> stan::math::log_falling_factorial ( const fvar< T > &  x, const fvar< T > &  n  )

inline

Definition at line 14 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_falling_factorial ( double  x, const fvar< T > &  n  )

inline

Definition at line 21 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_falling_factorial ( const fvar< T > &  x, double  n  )

inline

Definition at line 27 of file log_falling_factorial.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::log_falling_factorial ( const T1  x, const T2  n  )

inline

Return the natural log of the falling factorial of the specified arguments.

\[ \mbox{log\_falling\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \ln (x)_n & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_falling\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_falling\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ -\Psi(n) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first argument
T2	type of second argument

Parameters

[in]	x	First argument
[in]	n	Second argument

Returns

log of falling factorial of arguments

Exceptions

std::domain_error

if the first argument is not positive

Definition at line 55 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( const var &  a, double  b  )

inline

Definition at line 61 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( const var &  a, const var &  b  )

inline

Definition at line 65 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( double  a, const var &  b  )

inline

Definition at line 69 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_inv_logit ( const fvar< T > &  x )

inline

Definition at line 13 of file log_inv_logit.hpp.

var stan::math::log_inv_logit ( const var &  u )

inline

Return the natural logarithm of the inverse logit of the specified argument.

Parameters

u

argument

Returns

log inverse logit of the argument

Definition at line 20 of file log_inv_logit.hpp.

double stan::math::log_inv_logit ( double  u )

inline

Returns the natural logarithm of the inverse logit of the specified argument.

\[ \mbox{log\_inv\_logit}(x) = \begin{cases} \ln\left(\frac{1}{1+\exp(-x)}\right)& \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_inv\_logit}(x)}{\partial x} = \begin{cases} \frac{1}{1+\exp(x)} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

u

argument

Returns

log of the inverse logit of argument

Definition at line 32 of file log_inv_logit.hpp.

template<typename T >

apply_scalar_unary<log_inv_logit_fun, T>::return_t stan::math::log_inv_logit ( const T &  x )

inline

Return the elementwise application of log_inv_logit() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise log_inv_logit of members of container

Definition at line 40 of file log_inv_logit.hpp.

double stan::math::log_inv_logit ( int  u )

inline

Returns the natural logarithm of the inverse logit of the specified argument.

Parameters

u

argument

Returns

log of the inverse logit of argument

Definition at line 47 of file log_inv_logit.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::log_inv_logit_diff ( const T1 &  x, const T2 &  y  )

inline

Returns the natural logarithm of the difference of the inverse logits of the specified arguments.

\[ \mathrm{log\_inv\_logit\_diff}(x,y) = \ln\left(\frac{1}{1+\exp(-x)}-\frac{1}{1+\exp(-y)}\right) \]

\[ \frac{\partial }{\partial x} = -\frac{e^x}{e^y-e^x}-\frac{e^x}{e^x+1} \]

\[ \frac{\partial }{\partial x} = -\frac{e^y}{e^x-e^y}-\frac{e^y}{e^y+1} \]

Template Parameters

T1	Type of x argument.
T2	Type of y argument.

Parameters

x	Argument.
y	Argument.

Returns

Result of log difference of inverse logits of arguments.

Definition at line 35 of file log_inv_logit_diff.hpp.

template<typename T >

fvar<T> stan::math::log_inv_logit_diff ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Returns fvar with the natural logarithm of the difference of the inverse logits of the specified arguments and its gradients.

\[ \mathrm{log\_inv\_logit\_diff}(x,y) = \ln\left(\frac{1}{1+\exp(-x)}-\frac{1}{1+\exp(-y)}\right) \]

\[ \frac{\partial }{\partial x} = -\frac{e^x}{e^y-e^x}-\frac{e^x}{e^x+1} \]

\[ \frac{\partial }{\partial x} = -\frac{e^y}{e^x-e^y}-\frac{e^y}{e^y+1} \]

Template Parameters

T1	Type of x argument.
T2	Type of y argument.

Parameters

x	Argument.
y	Argument.

Returns

Fvar with result of log difference of inverse logits of arguments and gradients.

Definition at line 39 of file log_inv_logit_diff.hpp.

template<typename T >

fvar<T> stan::math::log_inv_logit_diff ( const fvar< T > &  x, double  y  )

inline

Definition at line 47 of file log_inv_logit_diff.hpp.

template<typename T >

fvar<T> stan::math::log_inv_logit_diff ( double  x, const fvar< T > &  y  )

inline

Definition at line 53 of file log_inv_logit_diff.hpp.

var stan::math::log_inv_logit_diff ( const var &  a, double  b  )

inline

Definition at line 71 of file log_inv_logit_diff.hpp.

var stan::math::log_inv_logit_diff ( const var &  a, const var &  b  )

inline

Definition at line 75 of file log_inv_logit_diff.hpp.

var stan::math::log_inv_logit_diff ( double  a, const var &  b  )

inline

Definition at line 79 of file log_inv_logit_diff.hpp.

double stan::math::log_mix ( double  theta, double  lambda1, double  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \lambda_1 + (1 - \theta) \lambda_2 \right). \]

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 27 of file log_mix.hpp.

template<typename T_theta , typename T_lam >

return_type_t<T_theta, T_lam> stan::math::log_mix	(	const T_theta &	theta,
		const T_lam &	lambda
	)

Return the log mixture density with specified mixing proportions and log densities.

\[ \frac{\partial }{\partial p_x} \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) =\frac{e^{d_x}}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

\[ \frac{\partial }{\partial d_x} \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) =\frac{e^{d_x}p_x}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

Parameters

theta	vector of mixing proportions in [0, 1].
lambda	vector of log densities.

Returns

log mixture of densities in specified proportion

Definition at line 38 of file log_mix.hpp.

double stan::math::log_mix ( double  theta, double  lambda1, int  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 44 of file log_mix.hpp.

double stan::math::log_mix ( double  theta, int  lambda1, double  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 57 of file log_mix.hpp.

double stan::math::log_mix ( int  theta, double  lambda1, double  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 70 of file log_mix.hpp.

template<typename T_theta , typename T_lambda1 , typename T_lambda2 >

return_type_t<T_theta, T_lambda1, T_lambda2> stan::math::log_mix ( const T_theta &  theta, const T_lambda1 &  lambda1, const T_lambda2 &  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities and its derivative at each.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right). \]

\[ \frac{\partial}{\partial \theta} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\exp(\lambda_1) - \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_1} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_1)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_2} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

Template Parameters

T_theta	theta scalar type.
T_lambda1	lambda1 scalar type.
T_lambda2	lambda2 scalar type.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 77 of file log_mix.hpp.

double stan::math::log_mix ( double  theta, int  lambda1, int  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 83 of file log_mix.hpp.

double stan::math::log_mix ( int  theta, double  lambda1, int  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 97 of file log_mix.hpp.

template<typename T >

fvar<T> stan::math::log_mix ( const fvar< T > &  theta, const fvar< T > &  lambda1, const fvar< T > &  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities and its derivative at each.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right). \]

\[ \frac{\partial}{\partial \theta} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\exp(\lambda_1) - \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_1} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_1)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_2} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

Template Parameters

T	scalar type.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 99 of file log_mix.hpp.

double stan::math::log_mix ( int  theta, int  lambda1, int  lambda2  )

inline

Return the log mixture density with specified mixing proportion and log densities.

Parameters

[in]	theta	mixing proportion in [0, 1].
[in]	lambda1	first log density.
[in]	lambda2	second log density.

Returns

log mixture of densities in specified proportion

Definition at line 111 of file log_mix.hpp.

template<typename T_theta , typename T_lam , int R, int C>

return_type_t<T_theta, std::vector<Eigen::Matrix<T_lam, R, C> > > stan::math::log_mix	(	const T_theta &	theta,
		const std::vector< Eigen::Matrix< T_lam, R, C >> &	lambda
	)

Return the log mixture density given specified mixing proportions and array of log density vectors.

\[ \frac{\partial }{\partial p_x}\left[ \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right)+ \log\left(\exp^{\log\left(p_1\right)+f_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+f_n}\right)\right] =\frac{e^{d_x}}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m}+ \frac{e^{f_x}}{e^{f_1}p_1+\cdot\cdot\cdot+e^{f_m}p_m} \]

\[ \frac{\partial }{\partial d_x}\left[ \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) +\log\left(\exp^{\log\left(p_1\right)+f_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+f_n}\right)\right] =\frac{e^{d_x}p_x}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

Parameters

theta	vector of mixing proportions in [0, 1].
lambda	array containing vectors of log densities.

Returns

log mixture of densities in specified proportion

Definition at line 117 of file log_mix.hpp.

template<typename T , typename P , require_all_arithmetic_t< P > ...>

fvar<T> stan::math::log_mix ( const fvar< T > &  theta, const fvar< T > &  lambda1, P  lambda2  )

inline

Definition at line 119 of file log_mix.hpp.

template<typename T , typename P , require_all_arithmetic_t< P > ...>

fvar<T> stan::math::log_mix ( const fvar< T > &  theta, P  lambda1, const fvar< T > &  lambda2  )

inline

Definition at line 137 of file log_mix.hpp.

template<typename T , typename P , require_all_arithmetic_t< P > ...>

fvar<T> stan::math::log_mix ( P  theta, const fvar< T > &  lambda1, const fvar< T > &  lambda2  )

inline

Definition at line 155 of file log_mix.hpp.

template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > ...>

fvar<T> stan::math::log_mix ( const fvar< T > &  theta, P1  lambda1, P2  lambda2  )

inline

Definition at line 174 of file log_mix.hpp.

template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > ...>

fvar<T> stan::math::log_mix ( P1  theta, const fvar< T > &  lambda1, P2  lambda2  )

inline

Definition at line 190 of file log_mix.hpp.

template<typename T , typename P1 , typename P2 , require_all_arithmetic_t< P1, P2 > ...>

fvar<T> stan::math::log_mix ( P1  theta, P2  lambda1, const fvar< T > &  lambda2  )

inline

Definition at line 206 of file log_mix.hpp.

template<typename T_theta , typename T_lam >

return_type_t<T_theta, std::vector<std::vector<T_lam> > > stan::math::log_mix	(	const T_theta &	theta,
		const std::vector< std::vector< T_lam >> &	lambda
	)

Return the log mixture density given specified mixing proportions and array of log density arrays.

\[ \frac{\partial }{\partial p_x}\left[ \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right)+ \log\left(\exp^{\log\left(p_1\right)+f_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+f_n}\right)\right] =\frac{e^{d_x}}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m}+ \frac{e^{f_x}}{e^{f_1}p_1+\cdot\cdot\cdot+e^{f_m}p_m} \]

\[ \frac{\partial }{\partial d_x}\left[ \log\left(\exp^{\log\left(p_1\right)+d_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+d_n}\right) +\log\left(\exp^{\log\left(p_1\right)+f_1}+\cdot\cdot\cdot+ \exp^{\log\left(p_n\right)+f_n}\right)\right] =\frac{e^{d_x}p_x}{e^{d_1}p_1+\cdot\cdot\cdot+e^{d_m}p_m} \]

Parameters

theta	vector of mixing proportions in [0, 1].
lambda	array containing arrays of log densities.

Returns

log mixture of densities in specified proportion

Definition at line 212 of file log_mix.hpp.

void stan::math::log_mix_partial_helper ( double  theta_val, double  lambda1_val, double  lambda2_val, double &  one_m_exp_lam2_m_lam1, double &  one_m_t_prod_exp_lam2_m_lam1, double &  one_d_t_plus_one_m_t_prod_exp_lam2_m_lam1  )

inline

Definition at line 23 of file log_mix.hpp.

template<typename T_theta , typename T_lambda1 , typename T_lambda2 , int N>

void stan::math::log_mix_partial_helper ( const T_theta &  theta, const T_lambda1 &  lambda1, const T_lambda2 &  lambda2, promote_args_t< T_theta, T_lambda1, T_lambda2 >(&)  partials_array[N]  )

inline

Definition at line 28 of file log_mix.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2, double> stan::math::log_modified_bessel_first_kind ( const T1  v, const T2  z  )

inline

Definition at line 38 of file log_modified_bessel_first_kind.hpp.

template<typename T >

fvar<T> stan::math::log_rising_factorial ( const fvar< T > &  x, const fvar< T > &  n  )

inline

Definition at line 13 of file log_rising_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_rising_factorial ( const fvar< T > &  x, double  n  )

inline

Definition at line 20 of file log_rising_factorial.hpp.

template<typename T >

fvar<T> stan::math::log_rising_factorial ( double  x, const fvar< T > &  n  )

inline

Definition at line 26 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( const var &  a, double  b  )

inline

Definition at line 43 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( const var &  a, const var &  b  )

inline

Definition at line 47 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( double  a, const var &  b  )

inline

Definition at line 51 of file log_rising_factorial.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::log_rising_factorial ( const T1 &  x, const T2 &  n  )

inline

Return the natural logarithm of the rising factorial from the first specified argument to the second.

\[ \mbox{log\_rising\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \ln x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_rising\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x+n) - \Psi(x) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_rising\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x+n) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first argument x
T2	type of second argument n

Parameters

[in]	x	first argument
[in]	n	second argument

Returns

natural logarithm of the rising factorial from x to n

Exceptions

std::domain_error

if the first argument is not positive

Definition at line 52 of file log_rising_factorial.hpp.

template<typename T >

Eigen::Matrix<fvar<T>, Eigen::Dynamic, 1> stan::math::log_softmax ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &  alpha )

inline

Definition at line 14 of file log_softmax.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::log_softmax ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v )

inline

Return the natural logarithm of the softmax of the specified vector.

\( \log \mbox{softmax}(y) \ = \ y - \log \sum_{k=1}^K \exp(y_k) \ = \ y - \mbox{log\_sum\_exp}(y). \)

For the log softmax function, the entries in the Jacobian are \( \frac{\partial}{\partial y_m} \mbox{softmax}(y)[k] = \left\{ \begin{array}{ll} 1 - \mbox{softmax}(y)[m] & \mbox{ if } m = k, \mbox{ and} \\[6pt] \mbox{softmax}(y)[m] & \mbox{ if } m \neq k. \end{array} \right. \)

Template Parameters

T	type of elements in the vector

Parameters

[in]

v

Vector to transform.

Returns

Unit simplex result of the softmax transform of the vector.

Definition at line 40 of file log_softmax.hpp.

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::log_softmax ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > &  alpha )

inline

Return the softmax of the specified Eigen vector.

Softmax is guaranteed to return a simplex.

The gradient calculations are unfolded.

Parameters

alpha

Unconstrained input vector.

Returns

Softmax of the input.

Exceptions

std::domain_error

If the input vector is size 0.

Definition at line 57 of file log_softmax.hpp.

template<typename T , int R, int C>

fvar<T> stan::math::log_sum_exp

(

const Eigen::Matrix< fvar< T >, R, C > &

v

)

Definition at line 12 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::math::log_sum_exp

(

const std::vector< fvar< T > > &

v

)

Definition at line 13 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::math::log_sum_exp ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 14 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::math::log_sum_exp ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 22 of file log_sum_exp.hpp.

double stan::math::log_sum_exp ( const std::vector< double > &  x )

inline

Return the log of the sum of the exponentiated values of the specified sequence of values.

The function is defined as follows to prevent overflow in exponential calculations.

\(\log \sum_{n=1}^N \exp(x_n) = \max(x) + \log \sum_{n=1}^N \exp(x_n - \max(x))\).

Parameters

[in]

x

array of specified values

Returns

The log of the sum of the exponentiated vector values.

Definition at line 25 of file log_sum_exp.hpp.

template<int R, int C>

double stan::math::log_sum_exp

(

const Eigen::Matrix< double, R, C > &

x

)

Return the log of the sum of the exponentiated values of the specified matrix of values.

The matrix may be a full matrix, a vector, or a row vector.

The function is defined as follows to prevent overflow in exponential calculations.

\(\log \sum_{n=1}^N \exp(x_n) = \max(x) + \log \sum_{n=1}^N \exp(x_n - \max(x))\).

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

[in]

x

Matrix of specified values

Returns

The log of the sum of the exponentiated vector values.

Definition at line 30 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::math::log_sum_exp ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 31 of file log_sum_exp.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::log_sum_exp ( const T2 &  a, const T1 &  b  )

inline

Calculates the log sum of exponetials without overflow.

\(\log (\exp(a) + \exp(b)) = m + \log(\exp(a-m) + \exp(b-m))\),

where \(m = max(a, b)\).

\[ \mbox{log\_sum\_exp}(x, y) = \begin{cases} \ln(\exp(x)+\exp(y)) & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_sum\_exp}(x, y)}{\partial x} = \begin{cases} \frac{\exp(x)}{\exp(x)+\exp(y)} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_sum\_exp}(x, y)}{\partial y} = \begin{cases} \frac{\exp(y)}{\exp(x)+\exp(y)} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	the first variable
b	the second variable

Definition at line 47 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const std::vector< var > &  x )

inline

Returns the log sum of exponentials.

Definition at line 48 of file log_sum_exp.hpp.

template<int R, int C>

var stan::math::log_sum_exp ( const Eigen::Matrix< var, R, C > &  x )

inline

Returns the log sum of exponentials.

Parameters

x

matrix

Definition at line 51 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const var &  a, const var &  b  )

inline

Returns the log sum of exponentials.

Definition at line 54 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const var &  a, double  b  )

inline

Returns the log sum of exponentials.

Definition at line 60 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( double  a, const var &  b  )

inline

Returns the log sum of exponentials.

Definition at line 66 of file log_sum_exp.hpp.

template<typename T1 , typename T2 >

int stan::math::logical_and ( const T1  x1, const T2  x2  )

inline

The logical and function which returns 1 if both arguments are unequal to zero and 0 otherwise.

Equivalent to x1 != 0 && x2 != 0.

\[ \mbox{operator\&\&}(x, y) = \begin{cases} 0 & \mbox{if } x = 0 \textrm{ or } y=0 \\ 1 & \mbox{if } x, y \neq 0 \\[6pt] 1 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters

T1	Type of first argument.
T2	Type of second argument.

Parameters

x1	First argument
x2	Second argument

Returns

true if both x1 and x2 are not equal to 0.

Definition at line 30 of file logical_and.hpp.

template<typename T1 , typename T2 >

int stan::math::logical_eq ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is equal to the second.

Equivalent to x1 == x2.

Template Parameters

T1	Type of first argument.
T2	Type of second argument.

Parameters

x1	First argument
x2	Second argument

Returns

true iff x1 == x2

Definition at line 19 of file logical_eq.hpp.

template<typename T1 , typename T2 >

bool stan::math::logical_gt ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is strictly greater than the second.

Equivalent to x1 < x2.

Template Parameters

T1	Type of first argument.
T2	Type of second argument.

Parameters

x1	First argument
x2	Second argument

Returns

true if x1 > x2

Definition at line 18 of file logical_gt.hpp.

template<typename T1 , typename T2 >

bool stan::math::logical_gte ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is greater than or equal to the second.

Equivalent to x1 >= x2.

Template Parameters

T1	Type of first argument.
T2	Type of second argument.

Parameters

x1	First argument
x2	Second argument

Returns

true if x1 >= x2

Definition at line 18 of file logical_gte.hpp.

template<typename T1 , typename T2 >

bool stan::math::logical_lt ( T1  x1, T2  x2  )

inline

Return 1 if the first argument is strictly less than the second.

Equivalent to x1 < x2.

Template Parameters

T1	Type of first argument.
T2	Type of second argument.

Parameters

x1	First argument
x2	Second argument

Returns

true if x1 < x2

Definition at line 18 of file logical_lt.hpp.

template<typename T1 , typename T2 >

bool stan::math::logical_lte ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is less than or equal to the second.

Equivalent to x1 <= x2.

Template Parameters

T1	Type of first argument.
T2	Type of second argument.

Parameters

x1	First argument
x2	Second argument

Returns

true iff x1 <= x2

Definition at line 18 of file logical_lte.hpp.

template<typename T >

int stan::math::logical_negation ( const T &  x )

inline

The logical negation function which returns one if the input is equal to zero and zero otherwise.

Template Parameters

T	type of value

Parameters

x

value

Returns

1 if value is zero and 0 otherwise

Definition at line 17 of file logical_negation.hpp.

template<typename T1 , typename T2 >

int stan::math::logical_neq ( const T1  x1, const T2  x2  )

inline

Return 1 if the first argument is unequal to the second.

Equivalent to x1 != x2.

Template Parameters

T1	Type of first argument.
T2	Type of second argument.

Parameters

x1	First argument
x2	Second argument

Returns

true iff x1 != x2

Definition at line 19 of file logical_neq.hpp.

template<typename T1 , typename T2 >

int stan::math::logical_or ( T1  x1, T2  x2  )

inline

The logical or function which returns 1 if either argument is unequal to zero and 0 otherwise.

Equivalent to x1 != 0 || x2 != 0.

\[ \mbox{operator||}(x, y) = \begin{cases} 0 & \mbox{if } x, y=0 \\ 1 & \mbox{if } x \neq 0 \textrm{ or } y\neq0\\[6pt] 1 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters

T1	Type of first argument.
T2	Type of second argument.

Parameters

x1	First argument
x2	Second argument

Returns

true if either x1 or x2 is not equal to 0.

Definition at line 29 of file logical_or.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 17 of file logistic_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 16 of file logistic_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 16 of file logistic_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 16 of file logistic_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::logistic_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 114 of file logistic_lpdf.hpp.

template<typename T >

fvar<T> stan::math::logit ( const fvar< T > &  x )

inline

Definition at line 14 of file logit.hpp.

var stan::math::logit ( const var &  u )

inline

Return the log odds of the specified argument.

Parameters

u

argument

Returns

log odds of argument

Definition at line 18 of file logit.hpp.

template<typename T >

apply_scalar_unary<logit_fun, T>::return_t stan::math::logit ( const T &  x )

inline

Return the elementwise application of logit() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise logit of container elements

Definition at line 38 of file logit.hpp.

double stan::math::logit ( double  u )

inline

Return the log odds of the argument.

The logit function is defined as for \(x \in [0, 1]\) by returning the log odds of \(x\) treated as a probability,

\(\mbox{logit}(x) = \log \left( \frac{x}{1 - x} \right)\).

The inverse to this function is inv_logit.

\[ \mbox{logit}(x) = \begin{cases} \textrm{NaN}& \mbox{if } x < 0 \textrm{ or } x > 1\\ \ln\frac{x}{1-x} & \mbox{if } 0\leq x \leq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{logit}(x)}{\partial x} = \begin{cases} \textrm{NaN}& \mbox{if } x < 0 \textrm{ or } x > 1\\ \frac{1}{x-x^2}& \mbox{if } 0\leq x\leq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

u

argument

Returns

log odds of argument

Definition at line 42 of file logit.hpp.

double stan::math::logit ( int  u )

inline

Return the log odds of the argument.

Parameters

u

argument

Returns

log odds of argument

Definition at line 53 of file logit.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 16 of file lognormal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 16 of file lognormal_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 16 of file lognormal_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 16 of file lognormal_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::lognormal_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 134 of file lognormal_lpdf.hpp.

template<typename T , typename L , typename U >

return_type_t<T, L, U> stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub  )

inline

Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds.

The transform is the transformed and scaled inverse logit,

\(f(x) = L + (U - L) \mbox{logit}^{-1}(x)\)

If the lower bound is negative infinity and upper bound finite, this function reduces to ub_constrain(x, ub). If the upper bound is positive infinity and the lower bound finite, this function reduces to lb_constrain(x, lb). If the upper bound is positive infinity and the lower bound negative infinity, this function reduces to identity_constrain(x).

Template Parameters

T	Type of scalar.
L	Type of lower bound.
U	Type of upper bound.

Parameters

[in]	x	Free scalar to transform.
[in]	lb	Lower bound.
[in]	ub	Upper bound.

Returns

Lower- and upper-bounded scalar derived from transforming the free scalar.

Exceptions

std::domain_error

if ub <= lb

Definition at line 44 of file lub_constrain.hpp.

template<typename T , typename L , typename U >

return_type_t<T, L, U> stan::math::lub_constrain ( const T &  x, const L &  lb, const U &  ub, T &  lp  )

inline

Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds and increment the specified log probability with the log absolute Jacobian determinant.

The transform is as defined in lub_constrain(T, double, double). The log absolute Jacobian determinant is given by

\(\log \left| \frac{d}{dx} \left( L + (U-L) \mbox{logit}^{-1}(x) \right) \right|\)

\( {} = \log | (U-L) \, (\mbox{logit}^{-1}(x)) \, (1 - \mbox{logit}^{-1}(x)) |\)

\( {} = \log (U - L) + \log (\mbox{logit}^{-1}(x)) + \log (1 - \mbox{logit}^{-1}(x))\)

If the lower bound is negative infinity and upper bound finite, this function reduces to ub_constrain(x, ub, lp). If the upper bound is positive infinity and the lower bound finite, this function reduces to lb_constrain(x, lb, lp). If the upper bound is positive infinity and the lower bound negative infinity, this function reduces to identity_constrain(x, lp).

Template Parameters

T	Type of scalar.
L	Type of lower bound.
U	Type of upper bound.

Parameters

[in]	x	Free scalar to transform.
[in]	lb	Lower bound.
[in]	ub	Upper bound.
[in,out]	lp	Log probability scalar reference.

Returns

Lower- and upper-bounded scalar derived from transforming the free scalar.

Exceptions

std::domain_error

if ub <= lb

Definition at line 114 of file lub_constrain.hpp.

template<typename T , typename L , typename U >

return_type_t<T, L, U> stan::math::lub_free ( const T &  y, const L &  lb, const U &  ub  )

inline

Return the unconstrained scalar that transforms to the specified lower- and upper-bounded scalar given the specified bounds.

The transfrom in lub_constrain(T, double, double), is reversed by a transformed and scaled logit,

\(f^{-1}(y) = \mbox{logit}(\frac{y - L}{U - L})\)

where \(U\) and \(L\) are the lower and upper bounds.

If the lower bound is negative infinity and upper bound finite, this function reduces to ub_free(y, ub). If the upper bound is positive infinity and the lower bound finite, this function reduces to lb_free(x, lb). If the upper bound is positive infinity and the lower bound negative infinity, this function reduces to identity_free(y).

Template Parameters

T	type of scalar
L	type of lower bound
U	type of upper bound

Parameters

y	constrained value
lb	lower bound
ub	upper bound

Returns

the free scalar that transforms to the input scalar given the bounds

Exceptions

std::invalid_argument

if the lower bound is greater than the upper bound, y is less than the lower bound, or y is greater than the upper bound

Definition at line 46 of file lub_free.hpp.

double stan::math::machine_precision ( )

inline

Returns the difference between 1.0 and the next value representable.

Returns

Minimum positive number.

Definition at line 128 of file constants.hpp.

template<typename T >

Eigen::Array<T, Eigen::Dynamic, 1> stan::math::make_nu	(	const T &	eta,
		size_t	K
	)

Return the degrees of freedom for the t distribution that corresponds to the shape parameter in the LKJ distribution.

Template Parameters

T	scalar type

Parameters

eta	LKJ distribution parameter in (0, inf)
K	number of variables in correlation matrix

Definition at line 19 of file make_nu.hpp.

template<int call_id, typename F , typename T_shared_param , typename T_job_param >

Eigen::Matrix<return_type_t<T_shared_param, T_job_param>, Eigen::Dynamic, 1> stan::math::map_rect	(	const Eigen::Matrix< T_shared_param, Eigen::Dynamic, 1 > &	shared_params,
		const std::vector< Eigen::Matrix< T_job_param, Eigen::Dynamic, 1 >> &	job_params,
		const std::vector< std::vector< double >> &	x_r,
		const std::vector< std::vector< int >> &	x_i,
		std::ostream *	msgs = nullptr
	)

Map N function evaluations to parameters and data which are in rectangular format.

Each function evaluation may return a column vector of different sizes and the output is the concatenated vector from all evaluations.

In addition to job specific parameters, real and int data, a shared parameter vector is repeated in all evaluations. All input parameters are stored as vectors whereas data is stored as arrays.

For N jobs the output of this function is

[ f(shared_params, job_params[1], x_r[1], x_i[1]), f(shared_params, job_params[2], x_r[2], x_i[2]), ... ]'.

The function is implemented with serial execution and with parallelism using threading or MPI (TODO). The threading version is used if the compiler flag STAN_THREADS is set during compilation while the MPI version is only available if STAN_MPI is defined. The MPI parallelism takes precedence over serial or threading execution of the function.

For the threaded parallelism the N jobs are chunked into T blocks which are executed asynchronously using the async C++11 facility. This ensure that at most T threads are used, but the actual number of threads is controlled by the implementation of async provided by the compiler. Note that nested calls of map_rect will lead to a multiplicative increase in the number of job chunks generated. The number of threads T is controlled at runtime via the STAN_NUM_threads environment variable, see the get_num_threads function for details.

For the MPI version to work this function has these special non-standard conventions:

• The call_id template parameter is considered as a label for the functor F and data combination. Since MPI communication is expensive, the real and int data is transmitted only a single time per functor F / call_id combination to the workers.
• The MPI implementation requires that the functor type fully specifies the functor and hence requires a default constructible function object. This choice reduces the need for communication across MPI as the type is sufficient and the state of the functor is not necessary to transmit. Thus, the functor is specified as template argument only.
• The size of the returned vector of each job must stay consistent when performing repeated calls to map_rect (which usually vary the values of the parameters).
• To achieve the exact same results between the serial and the MPI evaluation scheme both variants work in exactly the same way which is to use nested AD calls and pre-computing all gradients when the function gets called. The usual evaluation scheme would build an AD graph instead of doing on-the-spot gradient evaluation. For large problems this results in speedups for the serial version even on a single core due to smaller AD graph sizes.
• In MPI operation mode, no outputs from the workers are streamed back to the root.
• Finally, each map_rect call must be registered with the STAN_REGISTER_MAP_RECT macro. This is required to enable communication between processes for the MPI case. The macro definition is empty if MPI is not enabled.

The functor F is expected to have the usual operator() function with signature

template <typename T1, typename T2> Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, 1> operator()(const Eigen::Matrix<T1, Eigen::Dynamic, 1>& eta, const Eigen::Matrix<T2, Eigen::Dynamic, 1>& theta, const std::vector<double>& x_r, const std::vector<int>& x_i, std::ostream* msgs = 0) const { ... }

WARNING: For the MPI case, the data arguments are NOT checked if they are unchanged between repeated evaluations for a given call_id/functor F pair. This is silently assumed to be immutable between evaluations.

Template Parameters

T_shared_param	Type of shared parameters.
T_job_param	Type of job specific parameters.

Parameters

shared_params	shared parameter vector passed as first argument to functor for all jobs
job_params	Array of job specific parameter vectors. All job specific parameters must have matching sizes.
x_r	Array of real arrays for each job. The first dimension must match the number of jobs (which is determined by the first dimension of job_params) and each entry must have the same size.
x_i	Array of int data with the same conventions as x_r.
msgs	Output stream for messages.

Template Parameters

call_id	Label for functor/data combination. See above for details.
F	Functor which is applied to all job specific parameters with conventions described.

Returns

concatenated results from all jobs

Definition at line 123 of file map_rect.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::matrix_exp ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  A )

inline

Return the matrix exponential of the input matrix.

Template Parameters

T	type of elements in the matrix

Parameters

[in]

A

Matrix to exponentiate.

Returns

Matrix exponential, dynamically-sized.

Exceptions

<code>std::invalid_argument</code>

if the input matrix is not square.

Definition at line 22 of file matrix_exp.hpp.

template<typename T , int N>

Eigen::Matrix<T, N, N> stan::math::matrix_exp ( const Eigen::Matrix< T, N, N > &  A )

inline

Return the matrix exponential of the input statically-sized square matrix.

Template Parameters

T	type of elements in the matrix
N	size of the input square matrix.

Parameters

[in]

A

Matrix to exponentiate.

Returns

Matrix exponential, statically-sized.

Definition at line 46 of file matrix_exp.hpp.

template<typename T >

Eigen::Matrix<T, 1, 1> stan::math::matrix_exp ( const Eigen::Matrix< T, 1, 1 > &  A )

inline

Return the exponential of the input scalar when it's in the form of Eigen matrix.

Template Parameters

T	type of elements in the matrix

Returns

1x1 Matrix exponential, statically-sized.

Definition at line 66 of file matrix_exp.hpp.

template<typename Mtype >

Mtype stan::math::matrix_exp_2x2

(

const Mtype &

A

)

Return the matrix exponential of a 2x2 matrix.

Reference for algorithm: http://mathworld.wolfram.com/MatrixExponential.html Note: algorithm only works if delta > 0;

Template Parameters

Mtype

type of elements in the matrix

Parameters

[in]

A

2x2 matrix to exponentiate.

Returns

Matrix exponential of A.

Definition at line 21 of file matrix_exp_2x2.hpp.

template<int Cb>

Eigen::Matrix<double, -1, Cb> stan::math::matrix_exp_multiply ( const Eigen::MatrixXd &  A, const Eigen::Matrix< double,-1, Cb > &  B  )

inline

Return product of exp(A) and B, where A is a NxN double matrix, B is a NxCb double matrix, and t is a double.

Template Parameters

Cb	number of columns in matrix B, can be Eigen::Dynamic

Parameters

[in]	A	Matrix
[in]	B	Matrix

Returns

exponential of A multiplies B

Definition at line 20 of file matrix_exp_multiply.hpp.

template<typename Ta , typename Tb , int Cb>

Eigen::Matrix<typename stan::return_type<Ta, Tb>::type, -1, Cb> stan::math::matrix_exp_multiply ( const Eigen::Matrix< Ta,-1,-1 > &  A, const Eigen::Matrix< Tb,-1, Cb > &  B  )

inline

Wrapper of matrix_exp_action function for a more literal name.

Template Parameters

Ta	scalar type matrix A
Tb	scalar type matrix B
Cb	Columns matrix B

Parameters

[in]	A	Matrix
[in]	B	Matrix

Returns

exponential of A multiplies B

Definition at line 25 of file matrix_exp_multiply.hpp.

template<typename MatrixType >

MatrixType stan::math::matrix_exp_pade

(

const MatrixType &

arg

)

Computes the matrix exponential, using a Pade approximation, coupled with scaling and squaring.

Template Parameters

MatrixType

type of elements in the matrix

Parameters

[in]

arg

matrix to exponentiate.

Returns

Matrix exponential of input matrix.

Definition at line 20 of file matrix_exp_pade.hpp.

template<typename T_y , typename T_Mu , typename T_Sigma , typename T_D >

return_type_t<T_y, T_Mu, T_Sigma, T_D> stan::math::matrix_normal_prec_lpdf	(	const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &	y,
		const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &	Mu,
		const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &	Sigma,
		const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &	D
	)

Definition at line 83 of file matrix_normal_prec_lpdf.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::matrix_power ( const Eigen::Matrix< T, R, C > &  M, const int  n  )

inline

Returns the nth power of the specific matrix.

M^n = M * M * ... * M.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

[in]	M	a square matrix
[in]	n	exponent

Returns

nth power of M

Exceptions

std::domain_error	if the matrix contains NaNs or infinities.
std::invalid_argument	if the exponent is negative or the matrix is not square.

Definition at line 25 of file matrix_power.hpp.

template<int R, int C>

Eigen::Matrix<var, R, C> stan::math::matrix_power ( const Eigen::Matrix< var, R, C > &  M, const int  n  )

inline

Returns the nth power of the specific matrix.

M^n = M * M * ... * M.

Template Parameters

R	number of rows in matrix
C	number of columns in matrix

Parameters

[in]	M	a square matrix
[in]	n	exponent

Returns

nth power of M

Exceptions

std::domain_error	if the matrix contains NaNs or infinities.
std::invalid_argument	if the exponent is negative or the matrix is not square.

Definition at line 134 of file matrix_power.hpp.

int stan::math::max ( const std::vector< int > &  x )

inline

Returns the maximum coefficient in the specified column vector.

Parameters

x	specified vector

Returns

maximum coefficient value in the vector

Exceptions

<code>std::invalid_argument</code>

if the vector is size zero

Definition at line 21 of file max.hpp.

template<typename T >

T stan::math::max ( const std::vector< T > &  x )

inline

Returns the maximum coefficient in the specified column vector.

Template Parameters

type	of elements in the vector

Parameters

x	specified vector

Returns

maximum coefficient value in the vector, or -infinity if the vector is size zero

Definition at line 37 of file max.hpp.

template<typename T , int R, int C>

T stan::math::max ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the maximum coefficient in the specified matrix, vector, or row vector.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	specified matrix, vector, or row vector

Returns

maximum coefficient value in the vector, or -infinity if the vector is size zero

Definition at line 58 of file max.hpp.

template<typename T1 , typename T2 >

size_t stan::math::max_size	(	const T1 &	x1,
		const T2 &	x2
	)

Definition at line 10 of file max_size.hpp.

template<typename T1 , typename T2 , typename T3 >

size_t stan::math::max_size	(	const T1 &	x1,
		const T2 &	x2,
		const T3 &	x3
	)

Definition at line 17 of file max_size.hpp.

template<typename T1 , typename T2 , typename T3 , typename T4 >

size_t stan::math::max_size	(	const T1 &	x1,
		const T2 &	x2,
		const T3 &	x3,
		const T4 &	x4
	)

Definition at line 25 of file max_size.hpp.

template<typename T1 , typename T2 , typename T3 , typename T4 , typename T5 >

size_t stan::math::max_size	(	const T1 &	x1,
		const T2 &	x2,
		const T3 &	x3,
		const T4 &	x4,
		const T5 &	x5
	)

Definition at line 34 of file max_size.hpp.

template<typename T1 , typename T2 >

size_t stan::math::max_size_mvt	(	const T1 &	x1,
		const T2 &	x2
	)

Definition at line 11 of file max_size_mvt.hpp.

template<typename T1 , typename T2 , typename T3 >

size_t stan::math::max_size_mvt	(	const T1 &	x1,
		const T2 &	x2,
		const T3 &	x3
	)

Definition at line 18 of file max_size_mvt.hpp.

template<typename T1 , typename T2 , typename T3 , typename T4 >

size_t stan::math::max_size_mvt	(	const T1 &	x1,
		const T2 &	x2,
		const T3 &	x3,
		const T4 &	x4
	)

Definition at line 26 of file max_size_mvt.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A, const Eigen::Matrix< fvar< T >, R2, C2 > &  b  )

inline

Definition at line 18 of file mdivide_left.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< T1, R1, C1 > &  A, const Eigen::Matrix< T2, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b.

Template Parameters

T1	type of elements in first matrix
T2	type of elements in right-hand side matrix or vector
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C2	number of columns in the right-hand side matrix, can be Eigen::Dynamic

Parameters

A	Matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 30 of file mdivide_left.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< double, R1, C1 > &  A, const Eigen::Matrix< fvar< T >, R2, C2 > &  b  )

inline

Definition at line 57 of file mdivide_left.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Definition at line 77 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< var, R1, C1 > &  A, const Eigen::Matrix< var, R2, C2 > &  b  )

inline

Definition at line 176 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< var, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Definition at line 196 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< double, R1, C1 > &  A, const Eigen::Matrix< var, R2, C2 > &  b  )

inline

Definition at line 217 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2, typename T2 >

Eigen::Matrix<fvar<T2>, R1, C2> stan::math::mdivide_left_ldlt ( const LDLT_factor< double, R1, C1 > &  A, const Eigen::Matrix< fvar< T2 >, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 22 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2, typename T1 , typename T2 >

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_left_ldlt ( const LDLT_factor< T1, R1, C1 > &  A, const Eigen::Matrix< T2, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Template Parameters

T1	type of elements in the LDLT_factor
T2	type of elements in right-hand side matrix or vector
R1	number of rows in the LDLT_factor, can be Eigen::Dynamic
C1	number of columns in the LDLT_factor, can be Eigen::Dynamic
R2	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C2	number of columns in the right-hand side matrix, can be Eigen::Dynamic

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 31 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_ldlt ( const LDLT_factor< var, R1, C1 > &  A, const Eigen::Matrix< var, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 178 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_ldlt ( const LDLT_factor< var, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 203 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_ldlt ( const LDLT_factor< double, R1, C1 > &  A, const Eigen::Matrix< var, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 228 of file mdivide_left_ldlt.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_left_spd ( const Eigen::Matrix< T1, R1, C1 > &  A, const Eigen::Matrix< T2, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b where A is symmetric positive definite.

Template Parameters

T1	type of elements in the first matrix
T2	type of elements in the right-hand side matrix or vector
R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C2	number of columns in the right-hand side matrix, can be Eigen::Dynamic

Parameters

A	Matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 31 of file mdivide_left_spd.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_spd ( const Eigen::Matrix< var, R1, C1 > &  A, const Eigen::Matrix< var, R2, C2 > &  b  )

inline

Definition at line 144 of file mdivide_left_spd.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_spd ( const Eigen::Matrix< var, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Definition at line 163 of file mdivide_left_spd.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_spd ( const Eigen::Matrix< double, R1, C1 > &  A, const Eigen::Matrix< var, R2, C2 > &  b  )

inline

Definition at line 183 of file mdivide_left_spd.hpp.

template<int TriView, typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< T1, R1, C1 > &  A, const Eigen::Matrix< T2, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b when A is triangular.

Template Parameters

TriView	Specifies whether A is upper (Eigen::Upper) or lower triangular (Eigen::Lower).
T1	type of elements in the triangular matrix
T2	type of elements in the right-hand side matrix or vector
R1	number of rows in the triangular matrix, can be Eigen::Dynamic
C1	number of columns in the triangular matrix, can be Eigen::Dynamic
R2	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C2	number of columns in the right-hand side matrix, can be Eigen::Dynamic

Parameters

A	Triangular matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 35 of file mdivide_left_tri.hpp.

template<int TriView, typename T , int R1, int C1>

Eigen::Matrix<T, R1, C1> stan::math::mdivide_left_tri ( const Eigen::Matrix< T, R1, C1 > &  A )

inline

Returns the solution of the system Ax=b when A is triangular and b=I.

Template Parameters

T	type of elements in the matrix
R1	number of rows, can be Eigen::Dynamic
C1	number of columns, can be Eigen::Dynamic

Parameters

A	Triangular matrix.

Returns

x = A^-1 .

Exceptions

std::domain_error

if A is not square

Definition at line 59 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>

Eigen::Matrix<double, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< double, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Returns the solution of the system Ax=b when A is triangular and A and b are matrices of doubles.

Template Parameters

TriView	Specifies whether A is upper (Eigen::Upper) or lower triangular (Eigen::Lower).
T1	type of elements in the triangular matrix
T2	type of elements in the right-hand side matrix or vector
R1	number of rows in the triangular matrix, can be Eigen::Dynamic
C1	number of columns in the triangular matrix, can be Eigen::Dynamic
R2	number of rows in the right-hand side matrix, can be Eigen::Dynamic

Parameters

A	Triangular matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 89 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, int R1, int C1>

Eigen::Matrix<double, R1, C1> stan::math::mdivide_left_tri ( const Eigen::Matrix< double, R1, C1 > &  A )

inline

Returns the solution of the system Ax=b when A is triangular, b=I and both are matrices of doubles.

Template Parameters

TriView	Specifies whether A is upper (Eigen::Upper) or lower triangular (Eigen::Lower).
R1	number of rows, can be Eigen::Dynamic
C1	number of columns, can be Eigen::Dynamic

Parameters

A	Triangular matrix.

Returns

x = A^-1 .

Exceptions

std::domain_error

if A is not square

Definition at line 124 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< var, R1, C1 > &  A, const Eigen::Matrix< var, R2, C2 > &  b  )

inline

Definition at line 313 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< double, R1, C1 > &  A, const Eigen::Matrix< var, R2, C2 > &  b  )

inline

Definition at line 333 of file mdivide_left_tri.hpp.

template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>

Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< var, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Definition at line 354 of file mdivide_left_tri.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A, const Eigen::Matrix< fvar< T >, R2, C2 > &  b  )

inline

Definition at line 16 of file mdivide_left_tri_low.hpp.

template<typename T1 , typename T2 , typename = require_all_floating_point_t<T1, T2>>

matrix_cl<return_type_t<T1, T2> > stan::math::mdivide_left_tri_low ( const matrix_cl< T1 > &  A, const matrix_cl< T2 > &  b  )

inline

Returns the solution of the system Ax=b when A is lower triangular.

Template Parameters

T1	type of elements in A
T2	type of elements in b

Parameters

A	Triangular matrix.
b	Right hand side matrix or vector.

Returns

x = A^-1 b, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 24 of file mdivide_left_tri_low.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< T1, R1, C1 > &  A, const Eigen::Matrix< T2, R2, C2 > &  b  )

inline

Return the result of left dividing the second argument by the first argument.

Calling mdivide_left_tri_low(A, b) with divisor A and dividend b is more arithmetically stable than calling inv(A) * b.

Template Parameters

T1	type of elements in the divisor matrix
T2	type of elements in the dividend matrix
R1	number rows in the divisor matrix, can be Eigen::Dynamic
C1	number columns in the divisor matrix, can be Eigen::Dynamic
R2	number rows in the dividend matrix, can be Eigen::Dynamic
C2	number columns in the dividend matrix, can be Eigen::Dynamic

Parameters

A	divisor, an invertible square matrix
b	dividend, a matrix or vector with the same number of rows as the divisor has columns

Returns

left division of b by A

Exceptions

std::invalid_argument

if the divisor is not square or the dividend does not have the same number of rows as the divisor has columns.

Definition at line 34 of file mdivide_left_tri_low.hpp.

template<typename T , typename = require_all_floating_point_t<T>>

matrix_cl<T> stan::math::mdivide_left_tri_low ( const matrix_cl< T > &  A )

inline

Returns the solution of the system Ax=b when A is triangular and b=I.

Template Parameters

T	type of elements in A
R1	number of rows in A
C1	number of columns in A

Parameters

A	Triangular matrix.

Returns

x = A^-1 .

Exceptions

std::domain_error

if A is not square

Definition at line 41 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<T, R1, C1> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< T, R1, C1 > &  A )

inline

Definition at line 42 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< double, R1, C1 > &  A, const Eigen::Matrix< fvar< T >, R2, C2 > &  b  )

inline

Definition at line 57 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Definition at line 93 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A, const Eigen::Matrix< fvar< T >, R2, C2 > &  b  )

inline

Definition at line 18 of file mdivide_right.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_right ( const Eigen::Matrix< T1, R1, C1 > &  b, const Eigen::Matrix< T2, R2, C2 > &  A  )

inline

Returns the solution of the system xA=b.

Template Parameters

T1	type of elements in the right-hand side matrix or vector
T2	type of elements in the second matrix
R1	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C1	number of columns in the right-hand side matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

A	Matrix.
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 30 of file mdivide_right.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Definition at line 57 of file mdivide_right.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right ( const Eigen::Matrix< double, R1, C1 > &  A, const Eigen::Matrix< fvar< T >, R2, C2 > &  b  )

inline

Definition at line 78 of file mdivide_right.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_right_ldlt ( const Eigen::Matrix< T1, R1, C1 > &  b, const LDLT_factor< T2, R2, C2 > &  A  )

inline

Returns the solution of the system xA=b given an LDLT_factor of A.

Template Parameters

T1	type of elements in right-hand side matrix or vector
T2	type of elements in the LDLT_factor
R1	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C1	number of columns in the right-hand side matrix, can be Eigen::Dynamic
R2	number of rows in the LDLT_factor, can be Eigen::Dynamic
C2	number of columns in the LDLT_factor, can be Eigen::Dynamic

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if rows of b don't match the size of A.

Definition at line 32 of file mdivide_right_ldlt.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<double, R1, C2> stan::math::mdivide_right_ldlt ( const Eigen::Matrix< double, R1, C1 > &  b, const LDLT_factor< double, R2, C2 > &  A  )

inline

Definition at line 44 of file mdivide_right_ldlt.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_right_spd ( const Eigen::Matrix< T1, R1, C1 > &  b, const Eigen::Matrix< T2, R2, C2 > &  A  )

inline

Returns the solution of the system xA=b where A is symmetric positive definite.

Template Parameters

T1	type of elements in the right-hand side matrix or vector
T2	type of elements in the second matrix
R1	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C1	number of columns in the right-hand side matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

b	right-hand side matrix or vector
A	matrix

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 32 of file mdivide_right_spd.hpp.

template<Eigen::UpLoType TriView, typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_right_tri ( const Eigen::Matrix< T1, R1, C1 > &  b, const Eigen::Matrix< T2, R2, C2 > &  A  )

inline

Returns the solution of the system xA=b when A is triangular.

Template Parameters

TriView	Specifies whether A is upper (Eigen::Upper) or lower triangular (Eigen::Lower).
T1	type of elements in the right-hand side matrix or vector
T2	type of elements in the triangular matrix
R1	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C1	number of columns in the right-hand side matrix, can be Eigen::Dynamic
R2	number of rows in the triangular matrix, can be Eigen::Dynamic
C2	number of columns in the triangular matrix, can be Eigen::Dynamic

Parameters

A	Triangular matrix. Specify upper or lower with TriView being Eigen::Upper or Eigen::Lower.
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 36 of file mdivide_right_tri.hpp.

template<Eigen::UpLoType TriView, int R1, int C1, int R2, int C2>

Eigen::Matrix<double, R1, C2> stan::math::mdivide_right_tri ( const Eigen::Matrix< double, R1, C1 > &  b, const Eigen::Matrix< double, R2, C2 > &  A  )

inline

Returns the solution of the system xA=b when A is triangular and A and b are matrices of doubles.

Template Parameters

TriView	Specifies whether A is upper (Eigen::Upper) or lower triangular (Eigen::Lower).
R1	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C1	number of columns in the right-hand side matrix, can be Eigen::Dynamic
R2	number of rows in the triangular matrix, can be Eigen::Dynamic
C2	number of columns in the triangular matrix, can be Eigen::Dynamic

Parameters

A	Triangular matrix. Specify upper or lower with TriView being Eigen::Upper or Eigen::Lower.
b	Right hand side matrix or vector.

Returns

x = b A^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 77 of file mdivide_right_tri.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::mdivide_right_tri_low ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A, const Eigen::Matrix< fvar< T >, R2, C2 > &  b  )

inline

Definition at line 15 of file mdivide_right_tri_low.hpp.

template<typename T1 , typename T2 , typename = require_all_floating_point_t<T1, T2>>

matrix_cl<return_type_t<T1, T2> > stan::math::mdivide_right_tri_low ( const matrix_cl< T2 > &  b, const matrix_cl< T1 > &  A  )

inline

Returns the solution of the system Ax=b where A is a lower triangular matrix.

Parameters

A	Matrix.
b	Right hand side matrix or vector.

Returns

x = b * tri(A)^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 23 of file mdivide_right_tri_low.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::mdivide_right_tri_low ( const Eigen::Matrix< T1, R1, C1 > &  b, const Eigen::Matrix< T2, R2, C2 > &  A  )

inline

Returns the solution of the system x tri(A) = b when tri(A) is a lower triangular view of the matrix A.

Template Parameters

T1	type of elements in the right-hand side matrix or vector
T2	type of elements in the second matrix
R1	number of rows in the right-hand side matrix, can be Eigen::Dynamic
C1	number of columns in the right-hand side matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

b	right-hand side matrix or vector
A	matrix

Returns

x = b * tri(A)^-1, solution of the linear system.

Exceptions

std::domain_error

if A is not square or the rows of b don't match the size of A.

Definition at line 31 of file mdivide_right_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right_tri_low ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A, const Eigen::Matrix< double, R2, C2 > &  b  )

inline

Definition at line 56 of file mdivide_right_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right_tri_low ( const Eigen::Matrix< double, R1, C1 > &  A, const Eigen::Matrix< fvar< T >, R2, C2 > &  b  )

inline

Definition at line 85 of file mdivide_right_tri_low.hpp.

template<typename T >

return_type_t<T> stan::math::mean ( const std::vector< T > &  v )

inline

Returns the sample mean (i.e., average) of the coefficients in the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

v	Specified vector.

Returns

Sample mean of vector coefficients.

Exceptions

std::domain_error

if the size of the vector is less than 1.

Definition at line 22 of file mean.hpp.

template<typename T , int R, int C>

return_type_t<T> stan::math::mean ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the sample mean (i.e., average) of the coefficients in the specified vector, row vector, or matrix.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	Specified vector, row vector, or matrix.

Returns

Sample mean of vector coefficients.

Definition at line 40 of file mean.hpp.

int stan::math::min ( const std::vector< int > &  x )

inline

Returns the minimum coefficient in the specified column vector.

Parameters

x	specified vector

Returns

minimum coefficient value in the vector

Exceptions

<code>std::invalid_argument</code>

if the vector is size zero

Definition at line 21 of file min.hpp.

template<typename T >

T stan::math::min ( const std::vector< T > &  x )

inline

Returns the minimum coefficient in the specified column vector.

Template Parameters

T	type of elements in the vector

Parameters

x	specified vector

Returns

minimum coefficient value in the vector, or infinity if the vector is size zero

Definition at line 37 of file min.hpp.

template<typename T , int R, int C>

T stan::math::min ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the minimum coefficient in the specified matrix, vector, or row vector.

Template Parameters

T	type of elements in the matrix, vector or row vector
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	specified matrix, vector, or row vector

Returns

minimum coefficient value in the vector, or infinity if the vector is size zero

Definition at line 57 of file min.hpp.

template<typename T >

T stan::math::minus ( const T &  x )

inline

Returns the negation of the specified scalar or matrix.

Template Parameters

T	Type of subtrahend.

Parameters

x	Subtrahend.

Returns

Negation of subtrahend.

Definition at line 15 of file minus.hpp.

template<typename T >

fvar<T> stan::math::modified_bessel_first_kind ( int  v, const fvar< T > &  z  )

inline

Definition at line 12 of file modified_bessel_first_kind.hpp.

var stan::math::modified_bessel_first_kind ( int  v, const var &  a  )

inline

Definition at line 26 of file modified_bessel_first_kind.hpp.

template<typename T2 >

T2 stan::math::modified_bessel_first_kind ( int  v, const T2  z  )

inline

\[ \mbox{modified\_bessel\_first\_kind}(v, z) = \begin{cases} I_v(z) & \mbox{if } -\infty\leq z \leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{modified\_bessel\_first\_kind}(v, z)}{\partial z} = \begin{cases} \frac{\partial\, I_v(z)}{\partial z} & \mbox{if } -\infty\leq z\leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ {I_v}(z) = \left(\frac{1}{2}z\right)^v\sum_{k=0}^\infty \frac{\left(\frac{1}{4}z^2\right)^k}{k!\Gamma(v+k+1)} \]

\[ \frac{\partial \, I_v(z)}{\partial z} = I_{v-1}(z)-\frac{v}{z}I_v(z) \]

Definition at line 37 of file modified_bessel_first_kind.hpp.

template<typename T >

fvar<T> stan::math::modified_bessel_second_kind ( int  v, const fvar< T > &  z  )

inline

Definition at line 12 of file modified_bessel_second_kind.hpp.

var stan::math::modified_bessel_second_kind ( int  v, const var &  a  )

inline

Definition at line 26 of file modified_bessel_second_kind.hpp.

template<typename T2 >

T2 stan::math::modified_bessel_second_kind ( int  v, const T2  z  )

inline

\[ \mbox{modified\_bessel\_second\_kind}(v, z) = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ K_v(z) & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{modified\_bessel\_second\_kind}(v, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ \frac{\partial\, K_v(z)}{\partial z} & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ {K_v}(z) = \frac{\pi}{2}\cdot\frac{I_{-v}(z) - I_{v}(z)}{\sin(v\pi)} \]

\[ \frac{\partial \, K_v(z)}{\partial z} = -\frac{v}{z}K_v(z)-K_{v-1}(z) \]

Definition at line 42 of file modified_bessel_second_kind.hpp.

int stan::math::modulus ( int  x, int  y  )

inline

Definition at line 12 of file modulus.hpp.

template<typename T >

std::unique_lock< std::mutex > stan::math::mpi_broadcast_command

(

)

Broadcasts default constructible commands to the cluster.

Template Parameters

T	default constructible command class derived from mpi_command

Returns

A unique_lock instance locking the mpi_cluster

Definition at line 238 of file mpi_cluster.hpp.

std::unique_lock<std::mutex> stan::math::mpi_broadcast_command ( std::shared_ptr< mpi_command > &  command )

inline

Broadcasts a command instance to the listening cluster.

This function must be called on the root whenever the cluster is in listening mode and errs otherwise.

Parameters

command

shared pointer to an instance of a command class derived from mpi_command

Returns

A unique_lock instance locking the mpi_cluster

Definition at line 209 of file mpi_cluster.hpp.

std::vector<int> stan::math::mpi_map_chunks ( std::size_t  num_jobs, std::size_t  chunk_size = 1  )

inline

Maps jobs of given chunk size to workers and returning a vector of counts.

The returned vector is indexed by the rank of each worker and has size equal to the # of workers. Each count per worker is the product of the number of assigned jobs times the chunk size. The jobs are deterministically assigned to workers. This is used for static scheduling of jobs internally.

So with num_workers workers, then the counts for worker with given rank is

chunks[rank] = floor(num_jobs/num_workers) * chunk_size

The remainder jobs num_jobs % num_workers are assigned to rank >=1 workers such that the root (rank = 0) has a little less assigned chunks unless num_jobs < num_workes in which case the first num_jobs nodes recieve a job (including the root).

Parameters

num_jobs	Total number of jobs to dispatch
chunk_size	Chunk size per job

Returns

vector indexed by rank with the total number of elements mapped to a given worker

Definition at line 84 of file mpi_cluster.hpp.

template<typename T_y , typename T_covar , typename T_w >

return_type_t<T_y, T_covar, T_w> stan::math::multi_gp_cholesky_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  L, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &  w  )

inline

Definition at line 83 of file multi_gp_cholesky_lpdf.hpp.

template<typename T_y , typename T_covar , typename T_w >

return_type_t<T_y, T_covar, T_w> stan::math::multi_gp_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  Sigma, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &  w  )

inline

Definition at line 81 of file multi_gp_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_cholesky_lpdf ( const T_y &  y, const T_loc &  mu, const T_covar &  L  )

inline

Definition at line 160 of file multi_normal_cholesky_lpdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_covar &	Sigma
	)

Definition at line 14 of file multi_normal_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_lpdf ( const T_y &  y, const T_loc &  mu, const T_covar &  Sigma  )

inline

Definition at line 116 of file multi_normal_lpdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_prec_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_covar &	Sigma
	)

Definition at line 15 of file multi_normal_prec_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_covar >

return_type_t<T_y, T_loc, T_covar> stan::math::multi_normal_prec_lpdf ( const T_y &  y, const T_loc &  mu, const T_covar &  Sigma  )

inline

Definition at line 117 of file multi_normal_prec_lpdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::multi_student_t_lpdf ( const T_y &  y, const T_dof &  nu, const T_loc &  mu, const T_scale &  Sigma  )

inline

Definition at line 140 of file multi_student_t_lpdf.hpp.

template<bool propto, typename T_prob >

return_type_t<T_prob> stan::math::multinomial_lpmf	(	const std::vector< int > &	ns,
		const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &	theta
	)

Definition at line 15 of file multinomial_lpmf.hpp.

template<typename T_prob >

return_type_t<T_prob> stan::math::multinomial_lpmf	(	const std::vector< int > &	ns,
		const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &	theta
	)

Definition at line 45 of file multinomial_lpmf.hpp.

template<class RNG >

std::vector<int> stan::math::multinomial_rng ( const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  theta, int  N, RNG &  rng  )

inline

Definition at line 13 of file multinomial_rng.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const Eigen::Matrix< fvar< T >, R1, C1 > &  m, const fvar< T > &  c  )

inline

Definition at line 14 of file multiply.hpp.

template<typename T , int R2, int C2>

Eigen::Matrix<fvar<T>, R2, C2> stan::math::multiply ( const Eigen::Matrix< fvar< T >, R2, C2 > &  m, double  c  )

inline

Definition at line 23 of file multiply.hpp.

template<int R, int C, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::multiply ( const Eigen::Matrix< T1, R, C > &  m, T2  c  )

inline

Return specified matrix multiplied by specified scalar.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic
T1	type of elements in the matrix
T2	type of scalar

Parameters

m	matrix
c	scalar

Returns

product of matrix and scalar

Definition at line 29 of file multiply.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const Eigen::Matrix< double, R1, C1 > &  m, const fvar< T > &  c  )

inline

Definition at line 32 of file multiply.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const fvar< T > &  c, const Eigen::Matrix< fvar< T >, R1, C1 > &  m  )

inline

Definition at line 41 of file multiply.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( double  c, const Eigen::Matrix< fvar< T >, R1, C1 > &  m  )

inline

Definition at line 47 of file multiply.hpp.

template<int R, int C, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::multiply ( T1  c, const Eigen::Matrix< T2, R, C > &  m  )

inline

Return specified scalar multiplied by specified matrix.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic
T1	type of scalar
T2	type of elements in the matrix

Parameters

c	scalar
m	matrix

Returns

product of scalar and matrix

Definition at line 48 of file multiply.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const fvar< T > &  c, const Eigen::Matrix< double, R1, C1 > &  m  )

inline

Definition at line 53 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::multiply ( const Eigen::Matrix< fvar< T >, R1, C1 > &  m1, const Eigen::Matrix< fvar< T >, R2, C2 > &  m2  )

inline

Definition at line 59 of file multiply.hpp.

template<int R1, int C1, int R2, int C2, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

Eigen::Matrix<return_type_t<T1, T2>, R1, C2> stan::math::multiply ( const Eigen::Matrix< T1, R1, C1 > &  m1, const Eigen::Matrix< T2, R2, C2 > &  m2  )

inline

Return the product of the specified matrices.

The number of columns in the first matrix must be the same as the number of rows in the second matrix.

Template Parameters

R1	number of rows in the first matrix, can be Eigen::Dynamic
C1	number of columns in the first matrix, can be Eigen::Dynamic
R2	number of rows in the second matrix, can be Eigen::Dynamic
C2	number of columns in the second matrix, can be Eigen::Dynamic
T1	type of elements in first matrix
T2	type of elements in second matrix

Parameters

m1	first matrix
m2	second matrix

Returns

the product of the first and second matrices

Exceptions

<code>std::invalid_argument</code>

if the number of columns of m1 does not match the number of rows of m2.

Definition at line 73 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::multiply ( const Eigen::Matrix< fvar< T >, R1, C1 > &  m1, const Eigen::Matrix< double, R2, C2 > &  m2  )

inline

Definition at line 75 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, C2> stan::math::multiply ( const Eigen::Matrix< double, R1, C1 > &  m1, const Eigen::Matrix< fvar< T >, R2, C2 > &  m2  )

inline

Definition at line 91 of file multiply.hpp.

template<typename T , int C1, int R2>

fvar<T> stan::math::multiply ( const Eigen::Matrix< fvar< T >, 1, C1 > &  rv, const Eigen::Matrix< fvar< T >, R2, 1 > &  v  )

inline

Definition at line 107 of file multiply.hpp.

template<int C1, int R2, typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

return_type_t<T1, T2> stan::math::multiply ( const Eigen::Matrix< T1, 1, C1 > &  rv, const Eigen::Matrix< T2, R2, 1 > &  v  )

inline

Return the scalar product of the specified row vector and specified column vector.

The return is the same as the dot product. The two vectors must be the same size.

Template Parameters

C1	number of columns in row vector, can be Eigen::Dynamic
R2	number of rows in column vector, can be Eigen::Dynamic
T1	type of elements in row vector
T2	type of elements in column vector

Parameters

rv	row vector
v	column vector

Returns

scalar result of multiplying row vector by column vector

Exceptions

<code>std::invalid_argument</code>

if rv and v are not the same size

Definition at line 108 of file multiply.hpp.

template<typename T , int C1, int R2>

fvar<T> stan::math::multiply ( const Eigen::Matrix< fvar< T >, 1, C1 > &  rv, const Eigen::Matrix< double, R2, 1 > &  v  )

inline

Definition at line 114 of file multiply.hpp.

template<typename T , int C1, int R2>

fvar<T> stan::math::multiply ( const Eigen::Matrix< double, 1, C1 > &  rv, const Eigen::Matrix< fvar< T >, R2, 1 > &  v  )

inline

Definition at line 121 of file multiply.hpp.

template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>

return_type_t<T1, T2> stan::math::multiply ( T1  m, T2  c  )

inline

Return product of scalars.

Template Parameters

T1	type of first scalar
T2	type of second scalar

Parameters

m	scalar
c	scalar

Returns

product

Definition at line 124 of file multiply.hpp.

template<typename T1 , typename T2 , typename = require_all_var_or_arithmetic_t<T1, T2>, typename = require_any_var_t<T1, T2>>

return_type_t<T1, T2> stan::math::multiply ( const T1 &  v, const T2 &  c  )

inline

Return the product of two scalars.

Template Parameters

T1	scalar type of v
T2	scalar type of c

Parameters

[in]	v	First scalar
[in]	c	Specified scalar

Returns

Product of scalars

Definition at line 562 of file multiply.hpp.

template<int R, int C>

Eigen::Matrix<var, R, C> stan::math::multiply ( const var &  c, const Eigen::Matrix< var, R, C > &  m  )

inline

Return the product of scalar and matrix.

Template Parameters

R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	c	scalar of var type
[in]	m	matrix of var type

Returns

Product of scalar and matrix

Definition at line 575 of file multiply.hpp.

template<typename Arith , int R, int C, typename = require_arithmetic_t<Arith>>

Eigen::Matrix<var, R, C> stan::math::multiply ( const Arith &  c, const Eigen::Matrix< var, R, C > &  m  )

inline

Return the product of scalar and matrix.

Template Parameters

Arith	an arithmetic type
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	c	scalar of arithmetic type
[in]	m	matrix of var type

Returns

Product of scalar and matrix

Definition at line 590 of file multiply.hpp.

template<typename Arith , int R, int C, typename = require_arithmetic_t<Arith>>

Eigen::Matrix<var, R, C> stan::math::multiply ( const var &  c, const Eigen::Matrix< Arith, R, C > &  m  )

inline

Return the product of scalar and matrix.

Template Parameters

Arith	an arithmetic type
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	c	scalar of var type
[in]	m	matrix of arithmetic type

Returns

Product of scalar and matrix

Definition at line 605 of file multiply.hpp.

template<int R, int C>

Eigen::Matrix<var, R, C> stan::math::multiply ( const Eigen::Matrix< var, R, C > &  m, const var &  c  )

inline

Return the product of matrix and scalar.

Template Parameters

R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	m	matrix of var type
[in]	c	scalar of var type

Returns

Product of matrix and scalar

Definition at line 619 of file multiply.hpp.

template<typename Arith , int R, int C, typename = require_arithmetic_t<Arith>>

Eigen::Matrix<var, R, C> stan::math::multiply ( const Eigen::Matrix< Arith, R, C > &  m, const var &  c  )

inline

Return the product of matrix and scalar.

Template Parameters

Arith	an arithmetic type
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	m	matrix of arithmetic type
[in]	c	scalar of var type

Returns

Product of matrix and scalar

Definition at line 634 of file multiply.hpp.

template<typename Arith , int R, int C, typename = require_arithmetic_t<Arith>>

Eigen::Matrix<var, R, C> stan::math::multiply ( const Eigen::Matrix< var, R, C > &  m, const Arith &  c  )

inline

Return the product of matrix and scalar.

Template Parameters

Arith	an arithmetic type
R	number of rows or Eigen::Dynamic
C	number of columns or Eigen::Dynamic

Parameters

[in]	m	matrix of var type
[in]	c	scalar of arithmetic type

Returns

Product of matrix and scalar

Definition at line 649 of file multiply.hpp.

template<typename Ta , int Ra, int Ca, typename Tb , int Cb, typename = require_any_var_t<Ta, Tb>>

Eigen::Matrix<var, Ra, Cb> stan::math::multiply ( const Eigen::Matrix< Ta, Ra, Ca > &  A, const Eigen::Matrix< Tb, Ca, Cb > &  B  )

inline

Return the product of two matrices.

Template Parameters

Ta	scalar type matrix A
Ra	Rows matrix A
Ca	Columns matrix A
Tb	scalar type matrix B
RB	Rows matrix B
Cb	Columns matrix B

Parameters

[in]	A	Matrix
[in]	B	Matrix

Returns

Product of scalar and matrix.

Definition at line 668 of file multiply.hpp.

template<typename Ta , int Ca, typename Tb , typename = require_any_var_t<Ta, Tb>>

var stan::math::multiply ( const Eigen::Matrix< Ta, 1, Ca > &  A, const Eigen::Matrix< Tb, Ca, 1 > &  B  )

inline

Return the scalar product of a row vector and a vector.

Template Parameters

Ta	scalar type row vector A
Ca	Columns matrix A
Tb	scalar type vector B

Parameters

[in]	A	Row vector
[in]	B	Column vector

Returns

Scalar product of row vector and vector

Definition at line 696 of file multiply.hpp.

template<typename T >

fvar<T> stan::math::multiply_log ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 13 of file multiply_log.hpp.

template<typename T >

fvar<T> stan::math::multiply_log ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 20 of file multiply_log.hpp.

template<typename T >

fvar<T> stan::math::multiply_log ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 26 of file multiply_log.hpp.

template<typename T_a , typename T_b >

return_type_t<T_a, T_b> stan::math::multiply_log ( const T_a  a, const T_b  b  )

inline

Calculated the value of the first argument times log of the second argument while behaving properly with 0 inputs.

\( a * \log b \).

\[ \mbox{multiply\_log}(x, y) = \begin{cases} 0 & \mbox{if } x=y=0\\ x\ln y & \mbox{if } x, y\neq0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{multiply\_log}(x, y)}{\partial x} = \begin{cases} \infty & \mbox{if } x=y=0\\ \ln y & \mbox{if } x, y\neq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{multiply\_log}(x, y)}{\partial y} = \begin{cases} \infty & \mbox{if } x=y=0\\ \frac{x}{y} & \mbox{if } x, y\neq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	the first variable
b	the second variable

Returns

a * log(b)

Definition at line 51 of file multiply_log.hpp.

var stan::math::multiply_log ( const var &  a, const var &  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to a is log(b). The partial deriviative with respect to b is a/b. When a and b are both 0, this is set to Inf.

Parameters

a	First variable.
b	Second variable.

Returns

Value of a*log(b)

Definition at line 74 of file multiply_log.hpp.

var stan::math::multiply_log ( const var &  a, double  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to a is log(b).

Parameters

a	First variable.
b	Second scalar.

Returns

Value of a*log(b)

Definition at line 87 of file multiply_log.hpp.

var stan::math::multiply_log ( double  a, const var &  b  )

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to b is a/b. When a and b are both 0, this is set to Inf.

Parameters

a	First scalar.
b	Second variable.

Returns

Value of a*log(b)

Definition at line 101 of file multiply_log.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, R> stan::math::multiply_lower_tri_self_transpose ( const Eigen::Matrix< fvar< T >, R, C > &  m )

inline

Definition at line 13 of file multiply_lower_tri_self_transpose.hpp.

matrix_v stan::math::multiply_lower_tri_self_transpose ( const matrix_v &  L )

inline

Definition at line 14 of file multiply_lower_tri_self_transpose.hpp.

matrix_d stan::math::multiply_lower_tri_self_transpose ( const matrix_d &  L )

inline

Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose.

Parameters

L	Matrix to multiply.

Returns

The lower triangular values in L times their own transpose.

Exceptions

std::domain_error

If the input matrix is not square.

Definition at line 19 of file multiply_lower_tri_self_transpose.hpp.

template<typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_cdf	(	const T_n &	n,
		const T_location &	mu,
		const T_precision &	phi
	)

Definition at line 18 of file neg_binomial_2_cdf.hpp.

template<typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_lccdf	(	const T_n &	n,
		const T_location &	mu,
		const T_precision &	phi
	)

Definition at line 15 of file neg_binomial_2_lccdf.hpp.

template<typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_lcdf	(	const T_n &	n,
		const T_location &	mu,
		const T_precision &	phi
	)

Definition at line 14 of file neg_binomial_2_lcdf.hpp.

template<typename T_alpha , typename T_beta , typename T_precision >

return_type_t<T_alpha, T_beta, T_precision> stan::math::neg_binomial_2_log_glm_lpmf ( const matrix_cl< int > &  y, const matrix_cl< double > &  x, const T_alpha &  alpha, const T_beta &  beta, const T_precision &  phi  )

inline

Definition at line 201 of file neg_binomial_2_log_glm_lpmf.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_precision >

return_type_t<T_x, T_alpha, T_beta, T_precision> stan::math::neg_binomial_2_log_glm_lpmf ( const T_y &  y, const T_x &  x, const T_alpha &  alpha, const T_beta &  beta, const T_precision &  phi  )

inline

Definition at line 229 of file neg_binomial_2_log_glm_lpmf.hpp.

template<bool propto, typename T_n , typename T_log_location , typename T_precision >

return_type_t<T_log_location, T_precision> stan::math::neg_binomial_2_log_lpmf	(	const T_n &	n,
		const T_log_location &	eta,
		const T_precision &	phi
	)

Definition at line 20 of file neg_binomial_2_log_lpmf.hpp.

template<typename T_n , typename T_log_location , typename T_precision >

return_type_t<T_log_location, T_precision> stan::math::neg_binomial_2_log_lpmf ( const T_n &  n, const T_log_location &  eta, const T_precision &  phi  )

inline

Definition at line 116 of file neg_binomial_2_log_lpmf.hpp.

template<bool propto, typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_lpmf	(	const T_n &	n,
		const T_location &	mu,
		const T_precision &	phi
	)

Definition at line 19 of file neg_binomial_2_lpmf.hpp.

template<typename T_n , typename T_location , typename T_precision >

return_type_t<T_location, T_precision> stan::math::neg_binomial_2_lpmf ( const T_n &  n, const T_location &  mu, const T_precision &  phi  )

inline

Definition at line 115 of file neg_binomial_2_lpmf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_cdf	(	const T_n &	n,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 19 of file neg_binomial_cdf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_lccdf	(	const T_n &	n,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 20 of file neg_binomial_lccdf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_lcdf	(	const T_n &	n,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 20 of file neg_binomial_lcdf.hpp.

template<bool propto, typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_lpmf	(	const T_n &	n,
		const T_shape &	alpha,
		const T_inv_scale &	beta
	)

Definition at line 20 of file neg_binomial_lpmf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >

return_type_t<T_shape, T_inv_scale> stan::math::neg_binomial_lpmf ( const T_n &  n, const T_shape &  alpha, const T_inv_scale &  beta  )

inline

Definition at line 144 of file neg_binomial_lpmf.hpp.

double stan::math::negative_infinity ( )

inline

Return negative infinity.

Returns

Negative infinity.

Definition at line 113 of file constants.hpp.

static size_t stan::math::nested_size ( )

inlinestatic

Definition at line 10 of file nested_size.hpp.

template<typename T_alpha , typename T_beta , typename T_scale >

return_type_t<T_alpha, T_beta, T_scale> stan::math::normal_id_glm_lpdf ( const matrix_cl< double > &  y, const matrix_cl< double > &  x, const T_alpha &  alpha, const T_beta &  beta, const T_scale &  sigma  )

inline

Definition at line 198 of file normal_id_glm_lpdf.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta , typename T_scale >

return_type_t<T_y, T_x, T_alpha, T_beta, T_scale> stan::math::normal_id_glm_lpdf ( const T_y &  y, const T_x &  x, const T_alpha &  alpha, const T_beta &  beta, const T_scale &  sigma  )

inline

Definition at line 203 of file normal_id_glm_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::normal_lccdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 17 of file normal_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::normal_lcdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 20 of file normal_lcdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::normal_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 110 of file normal_lpdf.hpp.

template<typename T_y , typename T_s , typename T_n , typename T_loc , typename T_scale >

return_type_t<T_y, T_s, T_loc, T_scale> stan::math::normal_sufficient_lpdf ( const T_y &  y_bar, const T_s &  s_squared, const T_n &  n_obs, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 136 of file normal_sufficient_lpdf.hpp.

double stan::math::not_a_number ( )

inline

Return (quiet) not-a-number.

Returns

Quiet not-a-number.

Definition at line 120 of file constants.hpp.

template<typename T >

int stan::math::num_elements ( const T &  x )

inline

Returns 1, the number of elements in a primitive type.

Template Parameters

T	scalar type

Parameters

x	Argument of primitive type.

Returns

1

Definition at line 18 of file num_elements.hpp.

template<typename T , int R, int C>

int stan::math::num_elements ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the size of the specified matrix.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	argument matrix

Returns

size of matrix

Definition at line 33 of file num_elements.hpp.

template<typename T >

int stan::math::num_elements ( const std::vector< T > &  v )

inline

Returns the number of elements in the specified vector.

This assumes it is not ragged and that each of its contained elements has the same number of elements.

Template Parameters

T	type of elements in the vector

Parameters

v	argument vector

Returns

number of contained arguments

Definition at line 47 of file num_elements.hpp.

template<typename T , typename M , typename S >

return_type_t<T, M, S> stan::math::offset_multiplier_constrain ( const T &  x, const M &  mu, const S &  sigma  )

inline

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier.

The transform applied is

\(f(x) = mu + sigma * x\)

where mu is the offset and sigma is the multiplier.

If the offset is zero and the multiplier is one this reduces to identity_constrain(x).

Template Parameters

T	type of scalar
M	type of offset
S	type of multiplier

Parameters

[in]	x	Unconstrained scalar input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output

Returns

linear transformed value correspdonding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 38 of file offset_multiplier_constrain.hpp.

template<typename T , typename M , typename S >

return_type_t<T, M, S> stan::math::offset_multiplier_constrain ( const T &  x, const M &  mu, const S &  sigma, T &  lp  )

inline

Return the linearly transformed value for the specified unconstrained input and specified offset and multiplier, incrementing the specified reference with the log absolute Jacobian determinant of the transform.

The transform applied is

\(f(x) = mu + sigma * x\)

where mu is the offset and sigma is the multiplier.

If the offset is zero and multiplier is one, this function reduces to identity_constraint(x, lp).

Template Parameters

T	type of scalar
M	type of offset
S	type of multiplier

Parameters

[in]	x	Unconstrained scalar input
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output
[in,out]	lp	Reference to log probability to increment.

Returns

linear transformed value corresponding to inputs

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 79 of file offset_multiplier_constrain.hpp.

template<typename T , typename L , typename S >

return_type_t<T, L, S> stan::math::offset_multiplier_free ( const T &  y, const L &  mu, const S &  sigma  )

inline

Return the unconstrained scalar that transforms to the specified offset and multiplier constrained scalar given the specified offset and multiplier.

The transfrom in locmultiplier_constrain(T, double, double), is reversed by the reverse affine transformation,

\(f^{-1}(y) = \frac{y - L}{S}\)

where \(L\) and \(S\) are the offset and multiplier.

If the offset is zero and multiplier is one, this function reduces to identity_free(y).

Template Parameters

T	type of scalar
L	type of offset
S	type of multiplier

Parameters

	y	constrained value
[in]	mu	offset of constrained output
[in]	sigma	multiplier of constrained output

Returns

the free scalar that transforms to the input scalar given the offset and multiplier

Exceptions

std::domain_error	if sigma <= 0
std::domain_error	if mu is not finite

Definition at line 39 of file offset_multiplier_free.hpp.

bool stan::math::operator! ( const var &  x )

inline

Return the negation of the value of the argument as defined by !.

Parameters

[in]

x

argument

Returns

negation of argument value

Definition at line 16 of file operator_unary_not.hpp.

template<typename T >

bool stan::math::operator! ( const fvar< T > &  x )

inline

Return the negation of the value of the argument as defined by !.

Template Parameters

value

and tangent type for variables

Parameters

[in]

x

argument

Returns

negation of argument value

Definition at line 18 of file operator_unary_not.hpp.

template<typename T >

bool stan::math::operator!= ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the value of the two arguments are not equal as defined by !=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the values of the arguments are not equal

Definition at line 19 of file operator_not_equal.hpp.

bool stan::math::operator!= ( const var &  a, const var &  b  )

inline

Inequality operator comparing two variables' values (C++).

\[ \mbox{operator!=}(x, y) = \begin{cases} 0 & \mbox{if } x = y\\ 1 & \mbox{if } x \neq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if the first variable's value is not the same as the second's.

Definition at line 26 of file operator_not_equal.hpp.

template<typename T >

bool stan::math::operator!= ( const fvar< T > &  x, double  y  )

inline

Return true if the value of the first argument is not equal to the second argument as defined by !=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the value of the first argument is not equal to the second argument

Definition at line 34 of file operator_not_equal.hpp.

bool stan::math::operator!= ( const var &  a, double  b  )

inline

Inequality operator comparing a variable's value and a double (C++).

Parameters

a	First variable.
b	Second value.

Returns

True if the first variable's value is not the same as the second value.

Definition at line 39 of file operator_not_equal.hpp.

template<typename T >

bool stan::math::operator!= ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument is not equal to the value of the second argument as defined by !=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument is not equal to the value of the second argument

Definition at line 49 of file operator_not_equal.hpp.

bool stan::math::operator!= ( double  a, const var &  b  )

inline

Inequality operator comparing a double and a variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns

True if the first value is not the same as the second variable's value.

Definition at line 50 of file operator_not_equal.hpp.

bool stan::math::operator&& ( const var &  x, const var &  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjuntion of the argument's values

Definition at line 17 of file operator_logical_and.hpp.

template<typename T >

bool stan::math::operator&& ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjuntion of the argument's values

Definition at line 19 of file operator_logical_and.hpp.

template<typename T >

bool stan::math::operator&& ( const var &  x, double  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of first argument's value and second argument

Definition at line 31 of file operator_logical_and.hpp.

template<typename T >

bool stan::math::operator&& ( const fvar< T > &  x, double  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of first argument's value and second argument

Definition at line 34 of file operator_logical_and.hpp.

template<typename T >

bool stan::math::operator&& ( double  x, const var &  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of first argument and second argument's value

Definition at line 45 of file operator_logical_and.hpp.

template<typename T >

bool stan::math::operator&& ( double  x, const fvar< T > &  y  )

inline

Return the logical conjunction of the values of the two arguments as defined by &&.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

conjunction of first argument and the second argument's value

Definition at line 49 of file operator_logical_and.hpp.

template<typename T >

fvar<T> stan::math::operator* ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return the product of the two arguments.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

product of arguments

Definition at line 18 of file operator_multiplication.hpp.

template<typename T >

fvar<T> stan::math::operator* ( double  x, const fvar< T > &  y  )

inline

Return the product of the two arguments.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

product of arguments

Definition at line 31 of file operator_multiplication.hpp.

template<typename T >

fvar<T> stan::math::operator* ( const fvar< T > &  x, double  y  )

inline

Return the product of the two arguments.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

product of arguments

Definition at line 44 of file operator_multiplication.hpp.

var stan::math::operator* ( const var &  a, const var &  b  )

inline

Multiplication operator for two variables (C++).

The partial derivatives are

\(\frac{\partial}{\partial x} (x * y) = y\), and

\(\frac{\partial}{\partial y} (x * y) = x\).

\[ \mbox{operator*}(x, y) = \begin{cases} xy & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator*}(x, y)}{\partial x} = \begin{cases} y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator*}(x, y)}{\partial y} = \begin{cases} x & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of multiplying operands.

Definition at line 79 of file operator_multiplication.hpp.

var stan::math::operator* ( const var &  a, double  b  )

inline

Multiplication operator for a variable and a scalar (C++).

The partial derivative for the variable is

\(\frac{\partial}{\partial x} (x * c) = c\), and

Parameters

a	Variable operand.
b	Scalar operand.

Returns

Variable result of multiplying operands.

Definition at line 94 of file operator_multiplication.hpp.

var stan::math::operator* ( double  a, const var &  b  )

inline

Multiplication operator for a scalar and a variable (C++).

The partial derivative for the variable is

\(\frac{\partial}{\partial y} (c * y) = c\).

Parameters

a	Scalar operand.
b	Variable operand.

Returns

Variable result of multiplying the operands.

Definition at line 112 of file operator_multiplication.hpp.

template<typename T_a , typename T_b , typename = require_arithmetic_t<T_a>, typename = require_all_valid_expressions_t<T_b>>

elewise_multiplication_<scalar_<T_a>, as_operation_cl_t<T_b> > stan::math::operator* ( T_a &&  a, T_b &&  b  )

inline

Multiplication of a scalar and a kernel generator expression.

Template Parameters

T_a	type of scalar
T_b	type of expression

Parameters

a	scalar
b	expression

Returns

Multiplication of given arguments

Definition at line 212 of file binary_operation.hpp.

template<typename T_a , typename T_b , typename = require_all_valid_expressions_t<T_a>, typename = require_arithmetic_t<T_b>>

elewise_multiplication_<as_operation_cl_t<T_a>, scalar_<T_b> > stan::math::operator* ( T_a &&  a, const T_b  b  )

inline

Multiplication of a kernel generator expression and a scalar.

Template Parameters

T_a	type of expression
T_b	type of scalar

Parameters

a	expression
b	scalar

Returns

Multiplication of given arguments

Definition at line 229 of file binary_operation.hpp.

template<typename T_a , typename T_b , typename = require_all_valid_expressions_and_none_scalar_t<T_a, T_b>>

matrix_cl<double> stan::math::operator* ( const T_a &  a, const T_b &  b  )

inline

Matrix multiplication of two kernel generator expressions.

Evaluates both expressions before calculating the matrix product.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Matrix product of given arguments

Definition at line 245 of file binary_operation.hpp.

template<typename T >

fvar<T> stan::math::operator+ ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the sum of the specified forward mode addends.

Template Parameters

T	type of values and tangents

Parameters

x1	first addend
x2	second addend

Returns

sum of addends

Definition at line 18 of file operator_addition.hpp.

template<typename T >

fvar<T> stan::math::operator+ ( const fvar< T > &  x )

inline

Returns the argument.

It is included for completeness. The primitive unary operator+ exists to promote integer to floating point values.

Template Parameters

T	value and tangent type of the argument

Parameters

x

argument

Returns

the argument

Definition at line 19 of file operator_unary_plus.hpp.

template<typename T >

fvar<T> stan::math::operator+ ( double  x1, const fvar< T > &  x2  )

inline

Return the sum of the specified double and forward mode addends.

Template Parameters

T	type of values and tangents

Parameters

x1	first addend
x2	second addend

Returns

sum of addends

Definition at line 31 of file operator_addition.hpp.

var stan::math::operator+ ( const var &  a )

inline

Unary plus operator for variables (C++).

The function simply returns its input, because

\(\frac{d}{dx} +x = \frac{d}{dx} x = 1\).

The effect of unary plus on a built-in C++ scalar type is integer promotion. Because variables are all double-precision floating point already, promotion is not necessary.

\[ \mbox{operator+}(x) = \begin{cases} x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Argument variable.

Returns

The input reference.

Definition at line 43 of file operator_unary_plus.hpp.

template<typename T >

fvar<T> stan::math::operator+ ( const fvar< T > &  x1, double  x2  )

inline

Return the sum of the specified forward mode and double addends.

Template Parameters

T	type of values and tangents

Parameters

x1	first addend
x2	second addend

Returns

sum of addends

Definition at line 44 of file operator_addition.hpp.

var stan::math::operator+ ( const var &  a, const var &  b  )

inline

Addition operator for variables (C++).

The partial derivatives are defined by

\(\frac{\partial}{\partial x} (x+y) = 1\), and

\(\frac{\partial}{\partial y} (x+y) = 1\).

\[ \mbox{operator+}(x, y) = \begin{cases} x+y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x, y)}{\partial y} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of adding two variables.

Definition at line 80 of file operator_addition.hpp.

var stan::math::operator+ ( const var &  a, double  b  )

inline

Addition operator for variable and scalar (C++).

The derivative with respect to the variable is

\(\frac{d}{dx} (x + c) = 1\).

Parameters

a	First variable operand.
b	Second scalar operand.

Returns

Result of adding variable and scalar.

Definition at line 95 of file operator_addition.hpp.

var stan::math::operator+ ( double  a, const var &  b  )

inline

Addition operator for scalar and variable (C++).

The derivative with respect to the variable is

\(\frac{d}{dy} (c + y) = 1\).

Parameters

a	First scalar operand.
b	Second variable operand.

Returns

Result of adding variable and scalar.

Definition at line 113 of file operator_addition.hpp.

template<typename T_a , typename T_b , typename = require_all_valid_expressions_t<T_a, T_b>>

addition_<as_operation_cl_t<T_a>, as_operation_cl_t<T_b> > stan::math::operator+ ( T_a &&  a, T_b &&  b  )

inline

Addition of two kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first argument
b	second argument

Returns

Addition of given expressions

Definition at line 117 of file binary_operation.hpp.

var& stan::math::operator++ ( var &  a )

inline

Prefix increment operator for variables (C++).

Following C++, (++a) is defined to behave exactly as (a = a + 1.0) does, but is faster and uses less memory. In particular, the result is an assignable lvalue.

Parameters

a	Variable to increment.

Returns

Reference the result of incrementing this input variable.

Definition at line 35 of file operator_unary_increment.hpp.

var stan::math::operator++ ( var &  a, int    )

inline

Postfix increment operator for variables (C++).

Following C++, the expression (a++) is defined to behave like the sequence of operations

var temp = a; a = a + 1.0; return temp;

Parameters

a	Variable to increment.

Returns

Input variable.

Definition at line 51 of file operator_unary_increment.hpp.

template<typename T >

fvar<T> stan::math::operator- ( const fvar< T > &  x )

inline

Return the negation of the specified argument.

Template Parameters

value

and tangent type of the argument

Parameters

[in]

x

argument

Returns

negation of argument

Definition at line 17 of file operator_unary_minus.hpp.

template<typename T >

fvar<T> stan::math::operator- ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the difference of the specified arguments.

Template Parameters

T	type of values and tangents

Parameters

x1	first argument
x2	second argument

Returns

difference of the arguments

Definition at line 18 of file operator_subtraction.hpp.

template<typename T >

fvar<T> stan::math::operator- ( double  x1, const fvar< T > &  x2  )

inline

Return the difference of the specified arguments.

Template Parameters

T	type of values and tangents

Parameters

x1	first argument
x2	second argument

Returns

difference of the arguments

Definition at line 31 of file operator_subtraction.hpp.

template<typename T >

fvar<T> stan::math::operator- ( const fvar< T > &  x1, double  x2  )

inline

Return the difference of the specified arguments.

Template Parameters

T	type of values and tangents

Parameters

x1	first argument
x2	second argument

Returns

difference of the arguments

Definition at line 44 of file operator_subtraction.hpp.

var stan::math::operator- ( const var &  a )

inline

Unary negation operator for variables (C++).

\(\frac{d}{dx} -x = -1\).

\[ \mbox{operator-}(x) = \begin{cases} -x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Argument variable.

Returns

Negation of variable.

Definition at line 50 of file operator_unary_negative.hpp.

var stan::math::operator- ( const var &  a, const var &  b  )

inline

Subtraction operator for variables (C++).

The partial derivatives are defined by

\(\frac{\partial}{\partial x} (x-y) = 1\), and

\(\frac{\partial}{\partial y} (x-y) = -1\).

\[ \mbox{operator-}(x, y) = \begin{cases} x-y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x, y)}{\partial y} = \begin{cases} -1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of subtracting the second variable from the first.

Definition at line 93 of file operator_subtraction.hpp.

var stan::math::operator- ( const var &  a, double  b  )

inline

Subtraction operator for variable and scalar (C++).

The derivative for the variable is

\(\frac{\partial}{\partial x} (x-c) = 1\), and

Parameters

a	First variable operand.
b	Second scalar operand.

Returns

Result of subtracting the scalar from the variable.

Definition at line 108 of file operator_subtraction.hpp.

var stan::math::operator- ( double  a, const var &  b  )

inline

Subtraction operator for scalar and variable (C++).

The derivative for the variable is

\(\frac{\partial}{\partial y} (c-y) = -1\), and

Parameters

a	First scalar operand.
b	Second variable operand.

Returns

Result of sutracting a variable from a scalar.

Definition at line 126 of file operator_subtraction.hpp.

template<typename T_a , typename T_b , typename = require_all_valid_expressions_t<T_a, T_b>>

subtraction_<as_operation_cl_t<T_a>, as_operation_cl_t<T_b> > stan::math::operator- ( T_a &&  a, T_b &&  b  )

inline

Subtraction of two kernel generator expressions.

Template Parameters

T_a	type of first expression
T_b	type of second expression

Parameters

a	first expression
b	second expression

Returns

Subtraction of given expressions

Definition at line 151 of file binary_operation.hpp.

var& stan::math::operator-- ( var &  a )

inline

Prefix decrement operator for variables (C++).

Following C++, (–a) is defined to behave exactly as

a = a - 1.0)

does, but is faster and uses less memory. In particular, the result is an assignable lvalue.

Parameters

a	Variable to decrement.

Returns

Reference the result of decrementing this input variable.

Definition at line 39 of file operator_unary_decrement.hpp.

var stan::math::operator-- ( var &  a, int    )

inline

Postfix decrement operator for variables (C++).

Following C++, the expression (a–) is defined to behave like the sequence of operations

var temp = a; a = a - 1.0; return temp;

Parameters

a	Variable to decrement.

Returns

Input variable.

Definition at line 55 of file operator_unary_decrement.hpp.

template<typename T >

fvar<T> stan::math::operator/ ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return the result of dividing the first argument by the second.

Template Parameters

T	type of fvar value and tangent

Parameters

x1	first argument
x2	second argument

Returns

first argument divided by second argument

Definition at line 18 of file operator_division.hpp.

template<typename T >

fvar<T> stan::math::operator/ ( const fvar< T > &  x1, double  x2  )

inline

Return the result of dividing the first argument by the second.

Template Parameters

T	type of fvar value and tangent

Parameters

x1	first argument
x2	second argument

Returns

first argument divided by second argument

Definition at line 32 of file operator_division.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::operator/ ( const Eigen::Matrix< fvar< T >, R, C > &  v, const fvar< T > &  c  )

inline

Definition at line 38 of file divide.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::operator/ ( const Eigen::Matrix< fvar< T >, R, C > &  v, double  c  )

inline

Definition at line 44 of file divide.hpp.

template<typename T >

fvar<T> stan::math::operator/ ( double  x1, const fvar< T > &  x2  )

inline

Return the result of dividing the first argument by the second.

Template Parameters

T	type of fvar value and tangent

Parameters

x1	first argument
x2	second argument

Returns

first argument divided by second argument

Definition at line 45 of file operator_division.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::operator/ ( const Eigen::Matrix< double, R, C > &  v, const fvar< T > &  c  )

inline

Definition at line 50 of file divide.hpp.

var stan::math::operator/ ( const var &  a, const var &  b  )

inline

Division operator for two variables (C++).

The partial derivatives for the variables are

\(\frac{\partial}{\partial x} (x/y) = 1/y\), and

\(\frac{\partial}{\partial y} (x/y) = -x / y^2\).

\[ \mbox{operator/}(x, y) = \begin{cases} \frac{x}{y} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator/}(x, y)}{\partial x} = \begin{cases} \frac{1}{y} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator/}(x, y)}{\partial y} = \begin{cases} -\frac{x}{y^2} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable operand.
b	Second variable operand.

Returns

Variable result of dividing the first variable by the second.

Definition at line 88 of file operator_division.hpp.

var stan::math::operator/ ( const var &  a, double  b  )

inline

Division operator for dividing a variable by a scalar (C++).

The derivative with respect to the variable is

\(\frac{\partial}{\partial x} (x/c) = 1/c\).

Parameters

a	Variable operand.
b	Scalar operand.

Returns

Variable result of dividing the variable by the scalar.

Definition at line 103 of file operator_division.hpp.

var stan::math::operator/ ( double  a, const var &  b  )

inline

Division operator for dividing a scalar by a variable (C++).

The derivative with respect to the variable is

\(\frac{d}{d y} (c/y) = -c / y^2\).

Parameters

a	Scalar operand.
b	Variable operand.

Returns

Variable result of dividing the scalar by the variable.

Definition at line 121 of file operator_division.hpp.

template<typename T >

bool stan::math::operator< ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the first argument has a value less than the value of the second argument as defined by <.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument's value is less than the second argument's value

Definition at line 20 of file operator_less_than.hpp.

bool stan::math::operator< ( const var &  a, const var &  b  )

inline

Less than operator comparing variables' values (C++).

\[ \mbox{operator\textless}(x, y) = \begin{cases} 0 & \mbox{if } x \geq y \\ 1 & \mbox{if } x < y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if first variable's value is less than second's.

Definition at line 24 of file operator_less_than.hpp.

bool stan::math::operator< ( const var &  a, double  b  )

inline

Less than operator comparing variable's value and a double (C++).

Parameters

a	First variable.
b	Second value.

Returns

True if first variable's value is less than second value.

Definition at line 34 of file operator_less_than.hpp.

template<typename T >

bool stan::math::operator< ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument is less than the value of the second argument as defined by <.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument is less than the second's value argument

Definition at line 35 of file operator_less_than.hpp.

bool stan::math::operator< ( double  a, const var &  b  )

inline

Less than operator comparing a double and variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns

True if first value is less than second variable's value.

Definition at line 44 of file operator_less_than.hpp.

template<typename T >

bool stan::math::operator< ( const fvar< T > &  x, double  y  )

inline

Return true if the first argument has a value less than the second argument as defined by <.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a value less than the second argument argument

Definition at line 51 of file operator_less_than.hpp.

template<typename T >

bool stan::math::operator<= ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the first argument has a value less than or equal to the value of the second argument as defined by <=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument's value is less than or equal to the second argument's value

Definition at line 21 of file operator_less_than_or_equal.hpp.

bool stan::math::operator<= ( const var &  a, const var &  b  )

inline

Less than or equal operator comparing two variables' values (C++).

\[ \mbox{operator\textless=}(x, y) = \begin{cases} 0 & \mbox{if } x > y\\ 1 & \mbox{if } x \leq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if first variable's value is less than or equal to the second's.

Definition at line 26 of file operator_less_than_or_equal.hpp.

template<typename T >

bool stan::math::operator<= ( const fvar< T > &  x, double  y  )

inline

Return true if the first argument has a value less than or equal to the second argument as defined by <=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument's value is less than or equal to the second argument

Definition at line 36 of file operator_less_than_or_equal.hpp.

bool stan::math::operator<= ( const var &  a, double  b  )

inline

Less than or equal operator comparing a variable's value and a scalar (C++).

Parameters

a	First variable.
b	Second value.

Returns

True if first variable's value is less than or equal to the second value.

Definition at line 39 of file operator_less_than_or_equal.hpp.

bool stan::math::operator<= ( double  a, const var &  b  )

inline

Less than or equal operator comparing a double and variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns

True if first value is less than or equal to the second variable's value.

Definition at line 50 of file operator_less_than_or_equal.hpp.

template<typename T >

bool stan::math::operator<= ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument is less than or equal to the second argument's value as defined by <=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument is less than or equal to the second argument's value

Definition at line 51 of file operator_less_than_or_equal.hpp.

template<typename T >

bool stan::math::operator== ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the specified variables have equal values as defined by ==.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the arguments have equal values

Definition at line 19 of file operator_equal.hpp.

bool stan::math::operator== ( const var &  a, const var &  b  )

inline

Equality operator comparing two variables' values (C++).

\[ \mbox{operator==}(x, y) = \begin{cases} 0 & \mbox{if } x \neq y\\ 1 & \mbox{if } x = y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if the first variable's value is the same as the second's.

Definition at line 26 of file operator_equal.hpp.

template<typename T >

bool stan::math::operator== ( const fvar< T > &  x, double  y  )

inline

Return true if the the first variable has a value equal to the second argument as defined by by ==.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the arguments have equal values

Definition at line 34 of file operator_equal.hpp.

bool stan::math::operator== ( const var &  a, double  b  )

inline

Equality operator comparing a variable's value and a double (C++).

Parameters

a	First variable.
b	Second value.

Returns

True if the first variable's value is the same as the second value.

Definition at line 39 of file operator_equal.hpp.

template<typename T >

bool stan::math::operator== ( double  x, const fvar< T > &  y  )

inline

Return true if the the first argument is equal to the value of the second argument as defined by by ==.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the arguments have equal values

Definition at line 48 of file operator_equal.hpp.

bool stan::math::operator== ( double  a, const var &  b  )

inline

Equality operator comparing a scalar and a variable's value (C++).

Parameters

a	First scalar.
b	Second variable.

Returns

True if the variable's value is equal to the scalar.

Definition at line 49 of file operator_equal.hpp.

template<typename T >

bool stan::math::operator> ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the first argument has a greater value than the second as defined by >.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a greater value than the second

Definition at line 20 of file operator_greater_than.hpp.

bool stan::math::operator> ( const var &  a, const var &  b  )

inline

Greater than operator comparing variables' values (C++).

\[ \mbox{operator\textgreater}(x, y) = \begin{cases} 0 & \mbox{if } x \leq y\\ 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if first variable's value is greater than second's.

Definition at line 25 of file operator_greater_than.hpp.

bool stan::math::operator> ( const var &  a, double  b  )

inline

Greater than operator comparing variable's value and double (C++).

Parameters

a	First variable.
b	Second value.

Returns

True if first variable's value is greater than second value.

Definition at line 35 of file operator_greater_than.hpp.

template<typename T >

bool stan::math::operator> ( const fvar< T > &  x, double  y  )

inline

Return true if the first argument has a greater value than the second as defined by >.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a greater value than the second

Definition at line 35 of file operator_greater_than.hpp.

bool stan::math::operator> ( double  a, const var &  b  )

inline

Greater than operator comparing a double and a variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns

True if first value is greater than second variable's value.

Definition at line 45 of file operator_greater_than.hpp.

template<typename T >

bool stan::math::operator> ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument has a greater value than the second as defined by >.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a greater value than the second

Definition at line 50 of file operator_greater_than.hpp.

template<typename T >

bool stan::math::operator>= ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return true if the value of the first argument is greater than or equal to that of the second as defined by >=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a value greater than or equal to that of the second

Definition at line 20 of file operator_greater_than_or_equal.hpp.

bool stan::math::operator>= ( const var &  a, const var &  b  )

inline

Greater than or equal operator comparing two variables' values (C++).

\[ \mbox{operator\textgreater=}(x, y) = \begin{cases} 0 & \mbox{if } x < y\\ 1 & \mbox{if } x \geq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

a	First variable.
b	Second variable.

Returns

True if first variable's value is greater than or equal to the second's.

Definition at line 27 of file operator_greater_than_or_equal.hpp.

template<typename T >

bool stan::math::operator>= ( const fvar< T > &  x, double  y  )

inline

Return true if the value of the first argument has a value greater than or equal to the second argument as defined by >=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a value greater than or equal to that of the second

Definition at line 36 of file operator_greater_than_or_equal.hpp.

bool stan::math::operator>= ( const var &  a, double  b  )

inline

Greater than or equal operator comparing variable's value and double (C++).

Parameters

a	First variable.
b	Second value.

Returns

True if first variable's value is greater than or equal to second value.

Definition at line 40 of file operator_greater_than_or_equal.hpp.

bool stan::math::operator>= ( double  a, const var &  b  )

inline

Greater than or equal operator comparing double and variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns

True if the first value is greater than or equal to the second variable's value.

Definition at line 51 of file operator_greater_than_or_equal.hpp.

template<typename T >

bool stan::math::operator>= ( double  x, const fvar< T > &  y  )

inline

Return true if the first argument is greater than or equal to the value of the second argument as defined by >=.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

true if the first argument has a value greater than or equal to that of the second

Definition at line 52 of file operator_greater_than_or_equal.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::operator^ ( const Eigen::Matrix< T, R, C > &  M, const int  n  )

inline

Definition at line 49 of file matrix_power.hpp.

bool stan::math::operator|| ( const var &  x, const var &  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjuntion of the argument's values

Definition at line 17 of file operator_logical_or.hpp.

template<typename T >

bool stan::math::operator|| ( const fvar< T > &  x, const fvar< T > &  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjuntion of the argument's values

Definition at line 19 of file operator_logical_or.hpp.

template<typename T >

bool stan::math::operator|| ( const var &  x, double  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of first argument's value and second argument

Definition at line 31 of file operator_logical_or.hpp.

template<typename T >

bool stan::math::operator|| ( const fvar< T > &  x, double  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of first argument's value and second argument

Definition at line 34 of file operator_logical_or.hpp.

template<typename T >

bool stan::math::operator|| ( double  x, const var &  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of first argument and the second argument's value

Definition at line 45 of file operator_logical_or.hpp.

template<typename T >

bool stan::math::operator|| ( double  x, const fvar< T > &  y  )

inline

Return the logical disjunction of the values of the two arguments as defined by ||.

Template Parameters

value

and tangent type for variables

Parameters

[in]	x	first argument
[in]	y	second argument

Returns

disjunction of first argument and the second argument's value

Definition at line 49 of file operator_logical_or.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::ordered_constrain

(

const Eigen::Matrix< T, Eigen::Dynamic, 1 > &

x

)

Return an increasing ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of elements in the vector

Parameters

x	Free vector of scalars.

Returns

Positive, increasing ordered vector.

Template Parameters

T	Type of scalar.

Definition at line 22 of file ordered_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::ordered_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x, T &  lp  )

inline

Return a positive valued, increasing ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of elements in the vector

Parameters

x	Free vector of scalars.
lp	Log probability reference.

Returns

Positive, increasing ordered vector.

Definition at line 55 of file ordered_constrain.hpp.

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::ordered_constrain ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > &  x )

inline

Return an increasing ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Parameters

x	Free vector of scalars

Returns

Increasing ordered vector

Definition at line 89 of file ordered_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::ordered_free

(

const Eigen::Matrix< T, Eigen::Dynamic, 1 > &

y

)

Return the vector of unconstrained scalars that transform to the specified positive ordered vector.

This function inverts the constraining operation defined in ordered_constrain(Matrix),

Template Parameters

T	type of elements in the vector

Parameters

y	Vector of positive, ordered scalars.

Returns

Free vector that transforms into the input vector.

Exceptions

std::domain_error

if y is not a vector of positive, ordered scalars.

Definition at line 26 of file ordered_free.hpp.

template<typename T_beta_scalar , typename T_cuts_scalar >

return_type<T_beta_scalar, T_cuts_scalar>::type stan::math::ordered_logistic_glm_lpmf	(	const matrix_cl< int > &	y,
		const matrix_cl< double > &	x,
		const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, 1 > &	beta,
		const Eigen::Matrix< T_cuts_scalar, Eigen::Dynamic, 1 > &	cuts
	)

Definition at line 143 of file ordered_logistic_glm_lpmf.hpp.

template<typename T_y , typename T_x_scalar , int T_x_rows, typename T_beta_scalar , typename T_cuts_scalar >

stan::return_type_t<T_x_scalar, T_beta_scalar, T_cuts_scalar> stan::math::ordered_logistic_glm_lpmf	(	const T_y &	y,
		const Eigen::Matrix< T_x_scalar, T_x_rows, Eigen::Dynamic > &	x,
		const Eigen::Matrix< T_beta_scalar, Eigen::Dynamic, 1 > &	beta,
		const Eigen::Matrix< T_cuts_scalar, Eigen::Dynamic, 1 > &	cuts
	)

Definition at line 201 of file ordered_logistic_glm_lpmf.hpp.

template<typename T_y , typename T_loc , typename T_cut >

return_type_t<T_loc, T_cut> stan::math::ordered_logistic_lpmf	(	const T_y &	y,
		const T_loc &	lambda,
		const T_cut &	c
	)

Definition at line 167 of file ordered_logistic_lpmf.hpp.

template<class RNG >

int stan::math::ordered_logistic_rng ( double  eta, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  c, RNG &  rng  )

inline

Definition at line 14 of file ordered_logistic_rng.hpp.

template<typename T_loc , typename T_cut >

return_type_t<T_loc, T_cut> stan::math::ordered_probit_lpmf	(	int	y,
		const T_loc &	lambda,
		const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &	c
	)

Definition at line 64 of file ordered_probit_lpmf.hpp.

template<typename T_loc , typename T_cut >

return_type_t<T_loc, T_cut> stan::math::ordered_probit_lpmf	(	const std::vector< int > &	y,
		const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &	lambda,
		const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &	c
	)

Definition at line 131 of file ordered_probit_lpmf.hpp.

template<typename T_loc , typename T_cut >

return_type_t<T_loc, T_cut> stan::math::ordered_probit_lpmf	(	const std::vector< int > &	y,
		const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &	lambda,
		const std::vector< Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > > &	c
	)

Definition at line 208 of file ordered_probit_lpmf.hpp.

template<class RNG >

int stan::math::ordered_probit_rng ( double  eta, const Eigen::VectorXd &  c, RNG &  rng  )

inline

Definition at line 13 of file ordered_probit_rng.hpp.

void stan::math::out_of_range ( const char *  function, int  max, int  index, const char *  msg1 = "", const char *  msg2 = ""  )

inline

Throw an out_of_range exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing out of range. This will allow us to change the behavior for all functions at once. The message is: "<function>: index <index> out of range; expecting index to be between " "1 and <max><msg1><msg2>"

Parameters

function	Name of the function
max	Max
index	Index
msg1	Message to print. Default is "".
msg2	Message to print. Default is "".

Exceptions

std::out_of_range

with message.

Definition at line 27 of file out_of_range.hpp.

template<typename T >

fvar<T> stan::math::owens_t ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Return Owen's T function applied to the specified arguments.

Parameters

x1	First argument.
x2	Second argument.

Returns

Owen's T function applied to the specified arguments.

Definition at line 24 of file owens_t.hpp.

template<typename T >

fvar<T> stan::math::owens_t ( double  x1, const fvar< T > &  x2  )

inline

Return Owen's T function applied to the specified arguments.

Parameters

x1	First argument.
x2	Second argument.

Returns

Owen's T function applied to the specified arguments.

Definition at line 45 of file owens_t.hpp.

double stan::math::owens_t ( double  h, double  a  )

inline

Return the result of applying Owen's T function to the specified arguments.

Used to compute the cumulative density function for the skew normal distribution.

\[ \mbox{owens\_t}(h, a) = \begin{cases} \mbox{owens\_t}(h, a) & \mbox{if } -\infty\leq h, a \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial h} = \begin{cases} \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial h} & \mbox{if } -\infty\leq h, a\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial a} = \begin{cases} \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial a} & \mbox{if } -\infty\leq h, a\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \mbox{owens\_t}(h, a) = \frac{1}{2\pi} \int_0^a \frac{\exp(-\frac{1}{2}h^2(1+x^2))}{1+x^2}dx \]

\[ \frac{\partial \, \mbox{owens\_t}(h, a)}{\partial h} = -\frac{1}{2\sqrt{2\pi}} \operatorname{erf}\left(\frac{ha}{\sqrt{2}}\right) \exp\left(-\frac{h^2}{2}\right) \]

\[ \frac{\partial \, \mbox{owens\_t}(h, a)}{\partial a} = \frac{\exp\left(-\frac{1}{2}h^2(1+a^2)\right)}{2\pi (1+a^2)} \]

Parameters

h	First argument
a	Second argument

Returns

Owen's T function applied to the arguments.

Definition at line 57 of file owens_t.hpp.

template<typename T >

fvar<T> stan::math::owens_t ( const fvar< T > &  x1, double  x2  )

inline

Return Owen's T function applied to the specified arguments.

Parameters

x1	First argument.
x2	Second argument.

Returns

Owen's T function applied to the specified arguments.

Definition at line 63 of file owens_t.hpp.

var stan::math::owens_t ( const var &  h, const var &  a  )

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	var parameter.
a	var parameter.

Returns

The Owen's T function.

Definition at line 64 of file owens_t.hpp.

var stan::math::owens_t ( const var &  h, double  a  )

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	var parameter.
a	double parameter.

Returns

The Owen's T function.

Definition at line 78 of file owens_t.hpp.

var stan::math::owens_t ( double  h, const var &  a  )

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	double parameter.
a	var parameter.

Returns

The Owen's T function.

Definition at line 92 of file owens_t.hpp.

template<typename T , typename = require_var_t<T>>

std::vector<double> stan::math::packed_copy ( const matrix_cl< T > &  src )

inline

Packs the flat triangular matrix on the OpenCL device and copies the adjoint values to the std::vector.

Parameters

src	the flat triangular source matrix on the OpenCL device

Returns

the packed std::vector of adjoint values.

Exceptions

<code>std::invalid_argument</code>

if the matrix is not triangular

Definition at line 76 of file copy.hpp.

template<matrix_cl_view matrix_view>

matrix_cl<var> stan::math::packed_copy ( vari **  src, int  rows  )

inline

Copies the packed triangular matrix from the source std::vector to an OpenCL buffer and unpacks it to a flat matrix on the OpenCL device.

Template Parameters

matrix_view

the triangularity of the source matrix

Parameters

src	the packed source std::vector
rows	the number of rows in the flat matrix

Returns

the destination flat matrix on the OpenCL device

Exceptions

<code>std::invalid_argument</code>

if the size of the vector does not match the expected size for the packed triangular matrix

Definition at line 105 of file copy.hpp.

template<typename T_y , typename T_scale , typename T_shape >

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_cdf	(	const T_y &	y,
		const T_scale &	y_min,
		const T_shape &	alpha
	)

Definition at line 15 of file pareto_cdf.hpp.

template<typename T_y , typename T_scale , typename T_shape >

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_lccdf	(	const T_y &	y,
		const T_scale &	y_min,
		const T_shape &	alpha
	)

Definition at line 15 of file pareto_lccdf.hpp.

template<typename T_y , typename T_scale , typename T_shape >

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_lcdf	(	const T_y &	y,
		const T_scale &	y_min,
		const T_shape &	alpha
	)

Definition at line 15 of file pareto_lcdf.hpp.

template<bool propto, typename T_y , typename T_scale , typename T_shape >

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_lpdf	(	const T_y &	y,
		const T_scale &	y_min,
		const T_shape &	alpha
	)

Definition at line 16 of file pareto_lpdf.hpp.

template<typename T_y , typename T_scale , typename T_shape >

return_type_t<T_y, T_scale, T_shape> stan::math::pareto_lpdf ( const T_y &  y, const T_scale &  y_min, const T_shape &  alpha  )

inline

Definition at line 112 of file pareto_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	lambda,
		const T_shape &	alpha
	)

Definition at line 14 of file pareto_type_2_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	lambda,
		const T_shape &	alpha
	)

Definition at line 14 of file pareto_type_2_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	lambda,
		const T_shape &	alpha
	)

Definition at line 15 of file pareto_type_2_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	lambda,
		const T_shape &	alpha
	)

Definition at line 17 of file pareto_type_2_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::pareto_type_2_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  lambda, const T_shape &  alpha  )

inline

Definition at line 123 of file pareto_type_2_lpdf.hpp.

template<typename T , typename F >

void stan::math::partial_derivative	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		int	n,
		T &	fx,
		T &	dfx_dxn
	)

Return the partial derivative of the specified multiivariate function at the specified argument.

Template Parameters

T	Argument type
F	Function type

Parameters

	f	Function
[in]	x	Argument vector
[in]	n	Index of argument with which to take derivative
[out]	fx	Value of function applied to argument
[out]	dfx_dxn	Value of partial derivative

Definition at line 25 of file partial_derivative.hpp.

template<typename T >

fvar<T> stan::math::Phi ( const fvar< T > &  x )

inline

Definition at line 14 of file Phi.hpp.

double stan::math::Phi ( double  x )

inline

The unit normal cumulative distribution function.

The return value for a specified input is the probability that a random unit normal variate is less than or equal to the specified value, defined by

\(\Phi(x) = \int_{-\infty}^x \mbox{\sf Norm}(x|0, 1) \ dx\)

This function can be used to implement the inverse link function for probit regression.

Phi will underflow to 0 below -37.5 and overflow to 1 above 8

Parameters

x

Argument.

Returns

Probability random sample is less than or equal to argument.

Definition at line 30 of file Phi.hpp.

template<typename T >

apply_scalar_unary<Phi_fun, T>::return_t stan::math::Phi ( const T &  x )

inline

Vectorized version of Phi().

Template Parameters

T	type of container

Parameters

x

container

Returns

Unit normal CDF of each value in x.

Definition at line 32 of file Phi.hpp.

var stan::math::Phi ( const var &  a )

inline

The unit normal cumulative density function for variables (stan).

See Phi() for the double-based version.

The derivative is the unit normal density function,

\(\frac{d}{dx} \Phi(x) = \mbox{\sf Norm}(x|0, 1) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{1}{2} x^2)\).

\[ \mbox{Phi}(x) = \begin{cases} 0 & \mbox{if } x < -37.5 \\ \Phi(x) & \mbox{if } -37.5 \leq x \leq 8.25 \\ 1 & \mbox{if } x > 8.25 \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{Phi}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } x < -27.5 \\ \frac{\partial\, \Phi(x)}{\partial x} & \mbox{if } -27.5 \leq x \leq 27.5 \\ 0 & \mbox{if } x > 27.5 \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{0}^{x} e^{-t^2/2} dt \]

\[ \frac{\partial \, \Phi(x)}{\partial x} = \frac{e^{-x^2/2}}{\sqrt{2\pi}} \]

Parameters

a	Variable argument.

Returns

The unit normal cdf evaluated at the specified argument.

Definition at line 65 of file Phi.hpp.

double stan::math::Phi_approx ( double  x )

inline

Return an approximation of the unit normal CDF.

http://www.jiem.org/index.php/jiem/article/download/60/27

This function can be used to implement the inverse link function for probit regression.

Parameters

x

Argument.

Returns

Probability random sample is less than or equal to argument.

Definition at line 22 of file Phi_approx.hpp.

template<typename T >

fvar<T> stan::math::Phi_approx ( const fvar< T > &  x )

inline

Return an approximation of the unit normal cumulative distribution function (CDF).

Template Parameters

T	scalar type of forward-mode autodiff variable argument.

Parameters

x

argument

Returns

approximate probability random sample is less than or equal to argument

Definition at line 23 of file Phi_approx.hpp.

double stan::math::Phi_approx ( int  x )

inline

Return an approximation of the unit normal CDF.

Parameters

x

argument.

Returns

approximate probability random sample is less than or equal to argument.

Definition at line 34 of file Phi_approx.hpp.

template<typename T >

apply_scalar_unary<Phi_approx_fun, T>::return_t stan::math::Phi_approx ( const T &  x )

inline

Return the elementwise application of Phi_approx() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise Phi_approx of container elements

Definition at line 39 of file Phi_approx.hpp.

var stan::math::Phi_approx ( const var &  a )

inline

Approximation of the unit normal CDF for variables (stan).

http://www.jiem.org/index.php/jiem/article/download/60/27

\[ \mbox{Phi\_approx}(x) = \begin{cases} \Phi_{\mbox{\footnotesize approx}}(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{Phi\_approx}(x)}{\partial x} = \begin{cases} \frac{\partial\, \Phi_{\mbox{\footnotesize approx}}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Phi_{\mbox{\footnotesize approx}}(x) = \mbox{logit}^{-1}(0.07056 \, x^3 + 1.5976 \, x) \]

\[ \frac{\partial \, \Phi_{\mbox{\footnotesize approx}}(x)}{\partial x} = -\Phi_{\mbox{\footnotesize approx}}^2(x) e^{-0.07056x^3 - 1.5976x}(-0.21168x^2-1.5976) \]

Parameters

a	Variable argument.

Returns

The corresponding unit normal cdf approximation.

Definition at line 46 of file Phi_approx.hpp.

double stan::math::pi ( )

inline

Return the value of pi.

Returns

Pi.

Definition at line 78 of file constants.hpp.

template<typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_cdf	(	const T_n &	n,
		const T_rate &	lambda
	)

Definition at line 18 of file poisson_cdf.hpp.

template<typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_lccdf	(	const T_n &	n,
		const T_rate &	lambda
	)

Definition at line 18 of file poisson_lccdf.hpp.

template<typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_lcdf	(	const T_n &	n,
		const T_rate &	lambda
	)

Definition at line 18 of file poisson_lcdf.hpp.

template<typename T_alpha , typename T_beta >

return_type_t<T_alpha, T_beta> stan::math::poisson_log_glm_lpmf ( const matrix_cl< int > &  y, const matrix_cl< double > &  x, const T_alpha &  alpha, const T_beta &  beta  )

inline

Definition at line 144 of file poisson_log_glm_lpmf.hpp.

template<typename T_y , typename T_x , typename T_alpha , typename T_beta >

return_type_t<T_x, T_alpha, T_beta> stan::math::poisson_log_glm_lpmf ( const T_y &  y, const T_x &  x, const T_alpha &  alpha, const T_beta &  beta  )

inline

Definition at line 156 of file poisson_log_glm_lpmf.hpp.

template<bool propto, typename T_n , typename T_log_rate >

return_type_t<T_log_rate> stan::math::poisson_log_lpmf	(	const T_n &	n,
		const T_log_rate &	alpha
	)

Definition at line 17 of file poisson_log_lpmf.hpp.

template<typename T_n , typename T_log_rate >

return_type_t<T_log_rate> stan::math::poisson_log_lpmf ( const T_n &  n, const T_log_rate &  alpha  )

inline

Definition at line 83 of file poisson_log_lpmf.hpp.

template<bool propto, typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_lpmf	(	const T_n &	n,
		const T_rate &	lambda
	)

Definition at line 18 of file poisson_lpmf.hpp.

template<typename T_n , typename T_rate >

return_type_t<T_rate> stan::math::poisson_lpmf ( const T_n &  n, const T_rate &  lambda  )

inline

Definition at line 74 of file poisson_lpmf.hpp.

template<typename T >

T stan::math::positive_constrain ( const T &  x )

inline

Return the positive value for the specified unconstrained input.

The transform applied is

\(f(x) = \exp(x)\).

Parameters

x	Arbitrary input scalar.

Returns

Input transformed to be positive.

Definition at line 21 of file positive_constrain.hpp.

template<typename T >

T stan::math::positive_constrain ( const T &  x, T &  lp  )

inline

Return the positive value for the specified unconstrained input, incrementing the scalar reference with the log absolute Jacobian determinant.

See positive_constrain(T) for details of the transform. The log absolute Jacobian determinant is

\(\log | \frac{d}{dx} \mbox{exp}(x) | = \log | \mbox{exp}(x) | = x\).

Template Parameters

T	type of unconstrained value

Parameters

x	unconstrained value
lp	log density reference.

Returns

positive constrained version of unconstrained value

Definition at line 43 of file positive_constrain.hpp.

template<typename T >

T stan::math::positive_free ( const T &  y )

inline

Return the unconstrained value corresponding to the specified positive-constrained value.

The transform is the inverse of the transform \(f\) applied by positive_constrain(T), namely

\(f^{-1}(x) = \log(x)\).

The input is validated using check_positive().

Parameters

y	Input scalar.

Returns

Unconstrained value that produces the input when constrained.

Template Parameters

T	Type of scalar.

Exceptions

std::domain_error

if the variable is negative

Definition at line 28 of file positive_free.hpp.

double stan::math::positive_infinity ( )

inline

Return positive infinity.

Returns

Positive infinity.

Definition at line 106 of file constants.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::positive_ordered_constrain

(

const Eigen::Matrix< T, Eigen::Dynamic, 1 > &

x

)

Return an increasing positive ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of elements in the vector

Parameters

x	Free vector of scalars.

Returns

Positive, increasing ordered vector.

Definition at line 21 of file positive_ordered_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::positive_ordered_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x, T &  lp  )

inline

Return a positive valued, increasing positive ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The returned constrained vector will have the same dimensionality as the specified free vector.

Template Parameters

T	type of elements in the vector

Parameters

x	Free vector of scalars.
lp	Log probability reference.

Returns

Positive, increasing ordered vector.

Definition at line 53 of file positive_ordered_constrain.hpp.

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::positive_ordered_constrain ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > &  x )

inline

Return an increasing positive ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Parameters

x	Free vector of scalars

Returns

Positive, increasing ordered vector

Definition at line 86 of file positive_ordered_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::positive_ordered_free

(

const Eigen::Matrix< T, Eigen::Dynamic, 1 > &

y

)

Return the vector of unconstrained scalars that transform to the specified positive ordered vector.

This function inverts the constraining operation defined in positive_ordered_constrain(Matrix),

Template Parameters

T	type of elements in the vector

Parameters

y	Vector of positive, ordered scalars.

Returns

Free vector that transforms into the input vector.

Exceptions

std::domain_error

if y is not a vector of positive, ordered scalars.

Definition at line 26 of file positive_ordered_free.hpp.

template<typename T >

fvar<T> stan::math::pow ( const fvar< T > &  x1, const fvar< T > &  x2  )

inline

Definition at line 16 of file pow.hpp.

template<typename T >

fvar<T> stan::math::pow ( double  x1, const fvar< T > &  x2  )

inline

Definition at line 25 of file pow.hpp.

template<typename T >

fvar<T> stan::math::pow ( const fvar< T > &  x1, double  x2  )

inline

Definition at line 33 of file pow.hpp.

var stan::math::pow ( const var &  base, const var &  exponent  )

inline

Return the base raised to the power of the exponent (cmath).

The partial derivatives are

\(\frac{\partial}{\partial x} \mbox{pow}(x, y) = y x^{y-1}\), and

\(\frac{\partial}{\partial y} \mbox{pow}(x, y) = x^y \ \log x\).

\[ \mbox{pow}(x, y) = \begin{cases} x^y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{pow}(x, y)}{\partial x} = \begin{cases} yx^{y-1} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{pow}(x, y)}{\partial y} = \begin{cases} x^y\ln x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters

base	Base variable.
exponent	Exponent variable.

Returns

Base raised to the exponent.

Definition at line 108 of file pow.hpp.

var stan::math::pow ( const var &  base, double  exponent  )

inline

Return the base variable raised to the power of the exponent scalar (cmath).

The derivative for the variable is

\(\frac{d}{dx} \mbox{pow}(x, c) = c x^{c-1}\).

Parameters

base	Base variable.
exponent	Exponent scalar.

Returns

Base raised to the exponent.

Definition at line 124 of file pow.hpp.

var stan::math::pow ( double  base, const var &  exponent  )

inline

Return the base scalar raised to the power of the exponent variable (cmath).

The derivative for the variable is

\(\frac{d}{d y} \mbox{pow}(c, y) = c^y \log c \).

Parameters

base	Base scalar.
exponent	Exponent variable.

Returns

Base raised to the exponent.

Definition at line 158 of file pow.hpp.

var stan::math::precomputed_gradients ( double  value, const std::vector< var > &  operands, const std::vector< double > &  gradients  )

inline

This function returns a var for an expression that has the specified value, vector of operands, and vector of partial derivatives of value with respect to the operands.

Parameters

[in]	value	The value of the resulting dependent variable.
[in]	operands	operands.
[in]	gradients	vector of partial derivatives of result with respect to operands.

Returns

An auto-diff variable that uses the precomputed gradients provided.

Definition at line 90 of file precomputed_gradients.hpp.

double stan::math::primitive_value ( const var &  v )

inline

Return the primitive double value for the specified auto-diff variable.

Parameters

v	input variable.

Returns

value of input.

Definition at line 18 of file primitive_value.hpp.

template<typename T >

double stan::math::primitive_value ( const fvar< T > &  v )

inline

Return the primitive value of the specified forward-mode autodiff variable.

This function applies recursively to higher-order autodiff types to return a primitive double value.

Template Parameters

T	scalar type for autodiff variable.

Parameters

v	input variable.

Returns

primitive value of input.

Definition at line 21 of file primitive_value.hpp.

template<typename T >

std::enable_if<std::is_arithmetic<T>::value, T>::type stan::math::primitive_value ( T  x )

inline

Return the value of the specified arithmetic argument unmodified with its own declared type.

This template function can only be instantiated with arithmetic types as defined by std library's is_arithmetic trait metaprogram.

This function differs from value_of in that it does not cast all return types to double.

Template Parameters

T	type of arithmetic input.

Parameters

x

input.

Returns

input unmodified.

Definition at line 28 of file primitive_value.hpp.

template<typename T >

std::enable_if<!std::is_arithmetic<T>::value, double>::type stan::math::primitive_value ( const T &  x )

inline

Return the primitive value of the specified argument.

This implementation only applies to non-arithmetic types as defined by std libray's is_arithmetic trait metaprogram.

Template Parameters

T	type of non-arithmetic input.

Parameters

x

input.

Returns

value of input.

Definition at line 44 of file primitive_value.hpp.

template<int N>

void stan::math::print_mat_size ( std::ostream &  o )

inline

Helper function to return the matrix size as either "dynamic" or "1".

Template Parameters

N	Eigen matrix size specification

Parameters

o	output stream

Definition at line 22 of file assign.hpp.

void stan::math::print_stack ( std::ostream &  o )

inline

Prints the auto-dif variable stack.

This function is used for debugging purposes.

Only works if all members of stack are vari* as it casts to vari*.

Parameters

o	ostream to modify

Definition at line 20 of file print_stack.hpp.

template<typename T >

T stan::math::prob_constrain ( const T &  x )

inline

Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar.

The transform is the inverse logit,

\(f(x) = \mbox{logit}^{-1}(x) = \frac{1}{1 + \exp(x)}\).

Template Parameters

T	type of scalar

Parameters

[in]

x

unconstrained value

Returns

result constrained to fall in (0, 1)

Definition at line 25 of file prob_constrain.hpp.

template<typename T >

T stan::math::prob_constrain ( const T &  x, T &  lp  )

inline

Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The transform is as defined for prob_constrain(T). The log absolute Jacobian determinant is

The log absolute Jacobian determinant is

\(\log | \frac{d}{dx} \mbox{logit}^{-1}(x) |\)

\(\log ((\mbox{logit}^{-1}(x)) (1 - \mbox{logit}^{-1}(x))\)

\(\log (\mbox{logit}^{-1}(x)) + \log (1 - \mbox{logit}^{-1}(x))\).

Template Parameters

T	type of scalar

Parameters

[in]	x	unconstrained value
[in,out]	lp	log density

Returns

result constrained to fall in (0, 1)

Definition at line 50 of file prob_constrain.hpp.

template<typename T >

T stan::math::prob_free ( const T &  y )

inline

Return the free scalar that when transformed to a probability produces the specified scalar.

The function that reverses the constraining transform specified in prob_constrain(T) is the logit function,

\(f^{-1}(y) = \mbox{logit}(y) = \frac{1 - y}{y}\).

Template Parameters

T	type of constrained value

Parameters

y	constrained value

Returns

corresponding unconstrained value

Exceptions

std::domain_error

if y is not in (0, 1)

Definition at line 27 of file prob_free.hpp.

template<typename T >

T stan::math::prod ( const std::vector< T > &  v )

inline

Returns the product of the coefficients of the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

v	Specified vector.

Returns

Product of coefficients of vector.

Definition at line 19 of file prod.hpp.

template<typename T , int R, int C>

T stan::math::prod ( const Eigen::Matrix< T, R, C > &  v )

inline

Returns the product of the coefficients of the specified column vector.

Template Parameters

T	type of elements in the vector

Parameters

v	Specified vector.

Returns

Product of coefficients of vector.

Definition at line 36 of file prod.hpp.

template<typename T1 , typename T2 , typename F >

common_type<T1, T2>::type stan::math::promote_common ( const F &  u )

inline

Return the result of promoting either a scalar or the scalar elements of a container to either of two specified types, as determined by stan::math::common_type.

Template Parameters

T1	first type
T2	second type
F	type of container elements, must be either T1 or T2

Parameters

u	elements to promote

Returns

the result of promoting elements

Definition at line 23 of file promote_common.hpp.

template<typename T , typename S >

promote_scalar_type<T, S>::type stan::math::promote_scalar

(

const S &

x

)

This is the top-level function to call to promote the scalar types of an input of type S to type T.

Template Parameters

T	scalar type of output.
S	input type.

Parameters

x	input vector.

Returns

input vector with scalars promoted to type T.

Definition at line 62 of file promote_scalar.hpp.

template<typename T >

Eigen::Matrix<fvar<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_Q

(

const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Definition at line 12 of file qr_Q.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_Q

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Returns the orthogonal factor of the fat QR decomposition.

Template Parameters

T	type of elements in the matrix

Parameters

m

Matrix.

Returns

Orthogonal matrix with maximal columns

Definition at line 19 of file qr_Q.hpp.

template<typename T >

Eigen::Matrix<fvar<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_R

(

const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Definition at line 12 of file qr_R.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_R

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Returns the upper triangular factor of the fat QR decomposition.

Template Parameters

T	type of elements in the matrix

Parameters

m

Matrix.

Returns

Upper triangular matrix with maximal rows

Definition at line 19 of file qr_R.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_thin_Q

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Returns the orthogonal factor of the thin QR decomposition.

Template Parameters

T	type of elements in the matrix

Parameters

m

Matrix.

Returns

Orthogonal matrix with minimal columns

Definition at line 19 of file qr_thin_Q.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_thin_R

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Returns the upper triangular factor of the thin QR decomposition.

Template Parameters

T	type of elements in the matrix

Parameters

m

Matrix.

Returns

Upper triangular matrix with minimal rows

Definition at line 19 of file qr_thin_R.hpp.

template<int RA, int CA, int RB, int CB, typename T >

Eigen::Matrix<T, CB, CB> stan::math::quad_form ( const Eigen::Matrix< T, RA, CA > &  A, const Eigen::Matrix< T, RB, CB > &  B  )

inline

Compute B^T A B.

Definition at line 13 of file quad_form.hpp.

template<int RA, int CA, int RB, typename T >

T stan::math::quad_form ( const Eigen::Matrix< T, RA, CA > &  A, const Eigen::Matrix< T, RB, 1 > &  B  )

inline

Definition at line 21 of file quad_form.hpp.

template<typename Ta , int Ra, int Ca, typename Tb , int Rb, int Cb>

std::enable_if<std::is_same<Ta, var>::value || std::is_same<Tb, var>::value, Eigen::Matrix<var, Cb, Cb> >::type stan::math::quad_form ( const Eigen::Matrix< Ta, Ra, Ca > &  A, const Eigen::Matrix< Tb, Rb, Cb > &  B  )

inline

Definition at line 102 of file quad_form.hpp.

template<typename Ta , int Ra, int Ca, typename Tb , int Rb>

std::enable_if< std::is_same<Ta, var>::value || std::is_same<Tb, var>::value, var>::type stan::math::quad_form ( const Eigen::Matrix< Ta, Ra, Ca > &  A, const Eigen::Matrix< Tb, Rb, 1 > &  B  )

inline

Definition at line 116 of file quad_form.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, Eigen::Dynamic, Eigen::Dynamic> stan::math::quad_form_diag ( const Eigen::Matrix< T1, Eigen::Dynamic, Eigen::Dynamic > &  mat, const Eigen::Matrix< T2, R, C > &  vec  )

inline

Definition at line 12 of file quad_form_diag.hpp.

template<int RA, int CA, int RB, int CB, typename T >

Eigen::Matrix<T, CB, CB> stan::math::quad_form_sym ( const Eigen::Matrix< T, RA, CA > &  A, const Eigen::Matrix< T, RB, CB > &  B  )

inline

Definition at line 11 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, int CB, typename T >

Eigen::Matrix<fvar<T>, CB, CB> stan::math::quad_form_sym ( const Eigen::Matrix< fvar< T >, RA, CA > &  A, const Eigen::Matrix< double, RB, CB > &  B  )

inline

Definition at line 12 of file quad_form_sym.hpp.

template<typename Ta , int Ra, int Ca, typename Tb , int Rb, int Cb, require_any_var_t< Ta, Tb > ...>

Eigen::Matrix<var, Cb, Cb> stan::math::quad_form_sym ( const Eigen::Matrix< Ta, Ra, Ca > &  A, const Eigen::Matrix< Tb, Rb, Cb > &  B  )

inline

Definition at line 16 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, typename T >

T stan::math::quad_form_sym ( const Eigen::Matrix< T, RA, CA > &  A, const Eigen::Matrix< T, RB, 1 > &  B  )

inline

Definition at line 20 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, typename T >

fvar<T> stan::math::quad_form_sym ( const Eigen::Matrix< fvar< T >, RA, CA > &  A, const Eigen::Matrix< double, RB, 1 > &  B  )

inline

Definition at line 22 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, int CB, typename T >

Eigen::Matrix<fvar<T>, CB, CB> stan::math::quad_form_sym ( const Eigen::Matrix< double, RA, CA > &  A, const Eigen::Matrix< fvar< T >, RB, CB > &  B  )

inline

Definition at line 29 of file quad_form_sym.hpp.

template<typename Ta , int Ra, int Ca, typename Tb , int Rb, require_any_var_t< Ta, Tb > ...>

var stan::math::quad_form_sym ( const Eigen::Matrix< Ta, Ra, Ca > &  A, const Eigen::Matrix< Tb, Rb, 1 > &  B  )

inline

Definition at line 29 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, typename T >

fvar<T> stan::math::quad_form_sym ( const Eigen::Matrix< double, RA, CA > &  A, const Eigen::Matrix< fvar< T >, RB, 1 > &  B  )

inline

Definition at line 39 of file quad_form_sym.hpp.

template<typename C >

int stan::math::rank ( const C &  v, int  s  )

inline

Return the number of components of v less than v[s].

Template Parameters

C	container type

Parameters

[in]	v	input vector
[in]	s	position in vector

Returns

number of components of v less than v[s].

Exceptions

std::out_of_range

if s is out of range.

Definition at line 20 of file rank.hpp.

template<typename T_y , typename T_scale >

return_type_t<T_y, T_scale> stan::math::rayleigh_cdf	(	const T_y &	y,
		const T_scale &	sigma
	)

Definition at line 15 of file rayleigh_cdf.hpp.

template<typename T_y , typename T_scale >

return_type_t<T_y, T_scale> stan::math::rayleigh_lccdf	(	const T_y &	y,
		const T_scale &	sigma
	)

Definition at line 14 of file rayleigh_lccdf.hpp.

template<typename T_y , typename T_scale >

return_type_t<T_y, T_scale> stan::math::rayleigh_lcdf	(	const T_y &	y,
		const T_scale &	sigma
	)

Definition at line 15 of file rayleigh_lcdf.hpp.

template<bool propto, typename T_y , typename T_scale >

return_type_t<T_y, T_scale> stan::math::rayleigh_lpdf	(	const T_y &	y,
		const T_scale &	sigma
	)

Definition at line 15 of file rayleigh_lpdf.hpp.

template<typename T_y , typename T_scale >

return_type_t<T_y, T_scale> stan::math::rayleigh_lpdf ( const T_y &  y, const T_scale &  sigma  )

inline

Definition at line 80 of file rayleigh_lpdf.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_L	(	const Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs,
		size_t	K
	)

Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations.

It is generally better to work with the Cholesky factor rather than the correlation matrix itself when the determinant, inverse, etc. of the correlation matrix is needed for some statistical calculation.

See read_corr_matrix(Array, size_t, T) for more information.

Template Parameters

T	type of elements in the array

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.

Returns

Cholesky factor of correlation matrix for specified canonical partial correlations.

Definition at line 36 of file read_corr_L.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_L	(	const Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs,
		size_t	K,
		T &	log_prob
	)

Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation.

The implementation is Ben Goodrich's Cholesky factor-based approach to the C-vine method of:

• Daniel Lewandowski, Dorota Kurowicka, and Harry Joe, Generating random correlation matrices based on vines and extended onion method Journal of Multivariate Analysis 100 (2009) 1989–2001

Template Parameters

T	type of elements in the array

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.
log_prob	Reference to variable to increment with the log Jacobian determinant.

Returns

Cholesky factor of correlation matrix for specified partial correlations.

Definition at line 90 of file read_corr_L.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_matrix	(	const Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs,
		size_t	K
	)

Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations.

See read_corr_matrix(Array, size_t, T) for more information.

Template Parameters

T	type of elements in the array

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.

Returns

Cholesky factor of correlation matrix for specified canonical partial correlations.

Definition at line 25 of file read_corr_matrix.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_matrix	(	const Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs,
		size_t	K,
		T &	log_prob
	)

Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation.

It is usually preferable to utilize the version that returns the Cholesky factor of the correlation matrix rather than the correlation matrix itself in statistical calculations.

Template Parameters

T	type of elements in the array

Parameters

CPCs	The (K choose 2) canonical partial correlations in (-1, 1).
K	Dimensionality of correlation matrix.
log_prob	Reference to variable to increment with the log Jacobian determinant.

Returns

Correlation matrix for specified partial correlations.

Definition at line 53 of file read_corr_matrix.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_L	(	const Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs,
		const Eigen::Array< T, Eigen::Dynamic, 1 > &	sds,
		T &	log_prob
	)

This is the function that should be called prior to evaluating the density of any elliptical distribution.

Template Parameters

T	type of elements in the arrays

Parameters

CPCs	on (-1, 1)
sds	on (0, inf)
log_prob	the log probability value to increment with the Jacobian

Returns

Cholesky factor of covariance matrix for specified partial correlations.

Definition at line 22 of file read_cov_L.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_matrix	(	const Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs,
		const Eigen::Array< T, Eigen::Dynamic, 1 > &	sds,
		T &	log_prob
	)

A generally worse alternative to call prior to evaluating the density of an elliptical distribution.

Template Parameters

T	type of elements in the arrays

Parameters

CPCs	on (-1, 1)
sds	on (0, inf)
log_prob	the log probability value to increment with the Jacobian

Returns

Covariance matrix for specified partial correlations.

Definition at line 22 of file read_cov_matrix.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_matrix	(	const Eigen::Array< T, Eigen::Dynamic, 1 > &	CPCs,
		const Eigen::Array< T, Eigen::Dynamic, 1 > &	sds
	)

Builds a covariance matrix from CPCs and standard deviations.

Template Parameters

T	type of elements in the arrays

Parameters

CPCs	in (-1, 1)
sds	in (0, inf)

Definition at line 39 of file read_cov_matrix.hpp.

static void stan::math::recover_memory ( )

inlinestatic

Recover memory used for all variables for reuse.

Exceptions

std::logic_error

if empty_nested() returns false

Definition at line 18 of file recover_memory.hpp.

static void stan::math::recover_memory_nested ( )

inlinestatic

Recover only the memory used for the top nested call.

If there is nothing on the nested stack, then a std::logic_error exception is thrown.

Exceptions

std::logic_error

if empty_nested() returns true

Definition at line 20 of file recover_memory_nested.hpp.

template<typename T >

std::vector<T> stan::math::rep_array ( const T &  x, int  n  )

inline

Definition at line 12 of file rep_array.hpp.

template<typename T >

std::vector<std::vector<T> > stan::math::rep_array ( const T &  x, int  m, int  n  )

inline

Definition at line 18 of file rep_array.hpp.

template<typename T >

std::vector<std::vector<std::vector<T> > > stan::math::rep_array ( const T &  x, int  k, int  m, int  n  )

inline

Definition at line 26 of file rep_array.hpp.

template<typename T >

Eigen::Matrix<return_type_t<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::rep_matrix ( const T &  x, int  m, int  n  )

inline

Definition at line 12 of file rep_matrix.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::rep_matrix ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v, int  n  )

inline

Definition at line 20 of file rep_matrix.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::rep_matrix ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  rv, int  m  )

inline

Definition at line 29 of file rep_matrix.hpp.

template<typename T >

Eigen::Matrix<return_type_t<T>, 1, Eigen::Dynamic> stan::math::rep_row_vector ( const T &  x, int  m  )

inline

Definition at line 11 of file rep_row_vector.hpp.

template<typename T >

Eigen::Matrix<return_type_t<T>, Eigen::Dynamic, 1> stan::math::rep_vector ( const T &  x, int  n  )

inline

Definition at line 11 of file rep_vector.hpp.

template<typename T >

void stan::math::resize ( T &  x, std::vector< int >  dims  )

inline

Recursively resize the specified vector of vectors, which must bottom out at scalar values, Eigen vectors or Eigen matrices.

Template Parameters

T	type of object being resized

Parameters

x	Array-like object to resize.
dims	New dimensions.

Definition at line 43 of file resize.hpp.

var stan::math::rising_factorial ( const var &  a, int  b  )

inline

Definition at line 25 of file rising_factorial.hpp.

template<typename T >

fvar<T> stan::math::rising_factorial ( const fvar< T > &  x, int  n  )

inline

Return autodiff variable with the gradient and result of the rising factorial function applied to the inputs.

Will throw for NaN x and for negative n, as implemented in primitive function.

Template Parameters

T	Scalar type of autodiff variable.

Parameters

x	Argument.
n	Argument

Returns

tangent of rising factorial at arguments.

Definition at line 26 of file rising_factorial.hpp.

template<typename T >

return_type_t<T> stan::math::rising_factorial ( const T &  x, int  n  )

inline

Return the rising factorial function evaluated at the inputs.

Will throw for NaN x and for negative n

Template Parameters

T	Type of x argument.

Parameters

x	Argument.
n	Argument

Returns

Result of rising factorial function.

Exceptions

std::domain_error	if x is NaN
std::domain_error	if n is negative

\[ \mbox{rising\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{rising\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, x^{(n)}}{\partial x} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{rising\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, x^{(n)}}{\partial n} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)} \]

\[ \frac{\partial \, x^{(n)}}{\partial x} = x^{(n)}(\Psi(x+n)-\Psi(x)) \]

\[ \frac{\partial \, x^{(n)}}{\partial n} = (x)_n\Psi(x+n) \]

Definition at line 63 of file rising_factorial.hpp.

double stan::math::round ( double  x )

inline

Return the closest integer to the specified argument, with halfway cases rounded away from zero.

Parameters

x

Argument.

Returns

The rounded value of the argument.

Definition at line 17 of file round.hpp.

template<typename T >

fvar<T> stan::math::round ( const fvar< T > &  x )

inline

Return the closest integer to the specified argument, with halfway cases rounded away from zero.

The derivative is always zero.

Template Parameters

T	Scalar type for autodiff variable.

Parameters

x

Argument.

Returns

The rounded value of the argument.

Definition at line 24 of file round.hpp.

double stan::math::round ( int  x )

inline

Return the closest integer to the specified argument, with halfway cases rounded away from zero.

Parameters

x

Argument.

Returns

The rounded value of the argument.

Definition at line 26 of file round.hpp.

template<typename T >

apply_scalar_unary<round_fun, T>::return_t stan::math::round ( const T &  x )

inline

Vectorized version of round.

Template Parameters

T	type of container

Parameters

x

container

Returns

Rounded value of each value in x.

Definition at line 32 of file round.hpp.

var stan::math::round ( const var &  a )

inline

Returns the rounded form of the specified variable (C99).

The derivative is zero everywhere but numbers half way between whole numbers, so for convenience the derivative is defined to be everywhere zero,

\(\frac{d}{dx} \mbox{round}(x) = 0\).

\[ \mbox{round}(x) = \begin{cases} \lceil x \rceil & \mbox{if } x-\lfloor x\rfloor \geq 0.5 \\ \lfloor x \rfloor & \mbox{if } x-\lfloor x\rfloor < 0.5 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{round}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Specified variable.

Returns

Rounded variable.

Definition at line 55 of file round.hpp.

template<typename T >

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::row ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m, size_t  i  )

inline

Return the specified row of the specified matrix, using start-at-1 indexing.

This is equivalent to calling m.row(i - 1) and assigning the resulting template expression to a row vector.

Template Parameters

T	type of elements in the matrix

Parameters

m	Matrix.
i	Row index (count from 1).

Returns

Specified row of the matrix.

Exceptions

std::out_of_range

if i is out of range.

Definition at line 24 of file row.hpp.

template<typename T , int R, int C>

int stan::math::rows ( const Eigen::Matrix< T, R, C > &  m )

inline

Return the number of rows in the specified matrix, vector, or row vector.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

[in]

m

Input matrix, vector, or row vector.

Returns

Number of rows.

Definition at line 21 of file rows.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2  )

inline

Definition at line 13 of file rows_dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, require_any_var_t< T1, T2 > ...>

Eigen::Matrix<var, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< T1, R1, C1 > &  v1, const Eigen::Matrix< T2, R2, C2 > &  v2  )

inline

Definition at line 17 of file rows_dot_product.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<double, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Returns the dot product of the specified vectors.

Template Parameters

R1	number of rows in the first vector, can be Eigen::Dynamic
C1	number of columns in the first vector, can be Eigen::Dynamic
R2	number of rows in the second vector, can be Eigen::Dynamic
C2	number of columns in the second vector, can be Eigen::Dynamic

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 25 of file rows_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2  )

inline

Definition at line 27 of file rows_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Definition at line 41 of file rows_dot_product.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, 1> stan::math::rows_dot_self ( const Eigen::Matrix< fvar< T >, R, C > &  x )

inline

Definition at line 12 of file rows_dot_self.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, 1> stan::math::rows_dot_self ( const Eigen::Matrix< T, R, C > &  x )

inline

Returns the dot product of each row of a matrix with itself.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

x

matrix

Definition at line 19 of file rows_dot_self.hpp.

template<int Cb>

Eigen::Matrix<double, -1, Cb> stan::math::scale_matrix_exp_multiply ( const double &  t, const Eigen::MatrixXd &  A, const Eigen::Matrix< double,-1, Cb > &  B  )

inline

Return product of exp(At) and B, where A is a NxN double matrix, B is a NxCb double matrix, and t is a double.

Specialized for double values for efficiency.

Template Parameters

Cb	Columns matrix B

Parameters

[in]	A	Matrix
[in]	B	Matrix
[in]	t	double

Returns

exponential of At multiplies B

Definition at line 24 of file scale_matrix_exp_multiply.hpp.

template<typename Tt , typename Ta , typename Tb , int Cb>

Eigen::Matrix<stan::return_type_t<Tt, Ta, Tb>, -1, Cb> stan::math::scale_matrix_exp_multiply ( const Tt &  t, const Eigen::Matrix< Ta,-1,-1 > &  A, const Eigen::Matrix< Tb,-1, Cb > &  B  )

inline

Return product of exp(At) and B, where A is a NxN matrix, B is a NxCb matrix and t is a scalar.

Generic implementation when arguments are not double.

Template Parameters

Ta	scalar type matrix A
Tb	scalar type matrix B
Cb	Columns matrix B

Parameters

[in]	A	Matrix
[in]	B	Matrix
[in]	t	double

Returns

exponential of At multiplies B

Definition at line 53 of file scale_matrix_exp_multiply.hpp.

void stan::math::scaled_add ( std::vector< double > &  x, const std::vector< double > &  y, double  lambda  )

inline

Definition at line 11 of file scaled_add.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::scaled_inv_chi_square_lccdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_scale &	s
	)

Definition at line 19 of file scaled_inv_chi_square_lccdf.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::scaled_inv_chi_square_lcdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_scale &	s
	)

Definition at line 19 of file scaled_inv_chi_square_lcdf.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::scaled_inv_chi_square_lpdf ( const T_y &  y, const T_dof &  nu, const T_scale &  s  )

inline

Definition at line 159 of file scaled_inv_chi_square_lpdf.hpp.

template<typename T >

return_type_t<T> stan::math::sd ( const std::vector< T > &  v )

inline

Returns the unbiased sample standard deviation of the coefficients in the specified column vector.

Template Parameters

T	type of elements in the vector

Parameters

v	Specified vector.

Returns

Sample variance of vector.

Definition at line 23 of file sd.hpp.

template<typename T , int R, int C>

return_type_t<T> stan::math::sd ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the unbiased sample standard deviation of the coefficients in the specified vector, row vector, or matrix.

Template Parameters

T	type of elements in the vector, row vector, or matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m	Specified vector, row vector or matrix.

Returns

Sample variance.

Definition at line 43 of file sd.hpp.

var stan::math::sd ( const std::vector< var > &  v )

inline

Return the sample standard deviation of the specified standard vector.

Raise domain error if size is not greater than zero.

Parameters

[in]

v

a vector

Returns

sample standard deviation of specified vector

Definition at line 56 of file sd.hpp.

template<int R, int C>

var stan::math::sd

(

const Eigen::Matrix< var, R, C > &

m

)

Definition at line 75 of file sd.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::segment ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v, size_t  i, size_t  n  )

inline

Return the specified number of elements as a vector starting from the specified element - 1 of the specified vector.

Template Parameters

T	type of elements in the vector

Definition at line 18 of file segment.hpp.

template<typename T >

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::segment ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  v, size_t  i, size_t  n  )

inline

Definition at line 31 of file segment.hpp.

template<typename T >

std::vector<T> stan::math::segment	(	const std::vector< T > &	sv,
		size_t	i,
		size_t	n
	)

Definition at line 45 of file segment.hpp.

static void stan::math::set_zero_all_adjoints ( )

static

Reset all adjoint values in the stack to zero.

Definition at line 14 of file set_zero_all_adjoints.hpp.

static void stan::math::set_zero_all_adjoints_nested ( )

static

Reset all adjoint values in the top nested portion of the stack to zero.

Definition at line 17 of file set_zero_all_adjoints_nested.hpp.

template<typename T >

int stan::math::sign ( const T &  z )

inline

Definition at line 10 of file sign.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::simplex_constrain

(

const Eigen::Matrix< T, Eigen::Dynamic, 1 > &

y

)

Return the simplex corresponding to the specified free vector.

A simplex is a vector containing values greater than or equal to 0 that sum to 1. A vector with (K-1) unconstrained values will produce a simplex of size K.

The transform is based on a centered stick-breaking process.

Template Parameters

T	type of elements in the vector

Parameters

y	Free vector input of dimensionality K - 1.

Returns

Simplex of dimensionality K.

Definition at line 28 of file simplex_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::simplex_constrain	(	const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	y,
		T &	lp
	)

Return the simplex corresponding to the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The simplex transform is defined through a centered stick-breaking process.

Template Parameters

T	type of elements in the vector

Parameters

y	Free vector input of dimensionality K - 1.
lp	Log probability reference to increment.

Returns

Simplex of dimensionality K.

Definition at line 62 of file simplex_constrain.hpp.

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::simplex_constrain ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > &  y )

inline

Return the simplex corresponding to the specified free vector.

A simplex is a vector containing values greater than or equal to 0 that sum to 1. A vector with (K-1) unconstrained values will produce a simplex of size K.

The transform is based on a centered stick-breaking process.

Parameters

y	Free vector input of dimensionality K - 1

Returns

Simplex of dimensionality K

Definition at line 96 of file simplex_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::simplex_free

(

const Eigen::Matrix< T, Eigen::Dynamic, 1 > &

x

)

Return an unconstrained vector that when transformed produces the specified simplex.

It applies to a simplex of dimensionality K and produces an unconstrained vector of dimensionality (K-1).

The simplex transform is defined through a centered stick-breaking process.

Template Parameters

T	type of elements in the simplex

Parameters

x	Simplex of dimensionality K.

Returns

Free vector of dimensionality (K-1) that transforms to the simplex.

Exceptions

std::domain_error

if x is not a valid simplex

Definition at line 28 of file simplex_free.hpp.

template<typename T >

fvar<T> stan::math::sin ( const fvar< T > &  x )

inline

Definition at line 11 of file sin.hpp.

template<typename T >

apply_scalar_unary<sin_fun, T>::return_t stan::math::sin ( const T &  x )

inline

Vectorized version of sin().

Template Parameters

T	type of container

Parameters

x	angles in radians

Returns

Sine of each value in x.

Definition at line 33 of file sin.hpp.

var stan::math::sin ( const var &  a )

inline

Return the sine of a radian-scaled variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \sin x = \cos x\).

\[ \mbox{sin}(x) = \begin{cases} \sin(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sin}(x)}{\partial x} = \begin{cases} \cos(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable for radians of angle.

Returns

Sine of variable.

Definition at line 46 of file sin.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::singular_values

(

const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &

m

)

Return the vector of the singular values of the specified matrix in decreasing order of magnitude.

See the documentation for svd() for information on the singular values.

Template Parameters

T	type of elements in the matrix

Parameters

m	Specified matrix.

Returns

Singular values of the matrix.

Definition at line 20 of file singular_values.hpp.

template<typename T >

fvar<T> stan::math::sinh ( const fvar< T > &  x )

inline

Definition at line 11 of file sinh.hpp.

template<typename T >

apply_scalar_unary<sinh_fun, T>::return_t stan::math::sinh ( const T &  x )

inline

Vectorized version of sinh().

Template Parameters

T	type of container

Parameters

x

container

Returns

Hyperbolic sine of each variable in x.

Definition at line 33 of file sinh.hpp.

var stan::math::sinh ( const var &  a )

inline

Return the hyperbolic sine of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \sinh x = \cosh x\).

\[ \mbox{sinh}(x) = \begin{cases} \sinh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sinh}(x)}{\partial x} = \begin{cases} \cosh(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a

Variable.

Returns

Hyperbolic sine of variable.

Definition at line 46 of file sinh.hpp.

template<typename T , int R, int C>

size_t stan::math::size_mvt

(

const Eigen::Matrix< T, R, C > &

)

Definition at line 30 of file size_mvt.hpp.

template<typename T , int R, int C>

size_t stan::math::size_mvt

(

const std::vector< Eigen::Matrix< T, R, C > > &

x

)

Definition at line 35 of file size_mvt.hpp.

template<typename T >

bool stan::math::size_zero ( T &  x )

inline

Returns 1 if input is of length 0, returns 0 otherwise.

Parameters

x

argument

Returns

0 or 1

Definition at line 18 of file size_zero.hpp.

template<typename T , typename... Ts>

bool stan::math::size_zero ( T &  x, Ts &&...  xs  )

inline

Returns 1 if any inputs are of length 0, returns 0 otherwise.

Parameters

x	first argument
xs	parameter pack of remaining arguments to forward to function

Returns

0 or 1

Definition at line 31 of file size_zero.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_cdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	alpha
	)

Definition at line 18 of file skew_normal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_lccdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	alpha
	)

Definition at line 18 of file skew_normal_lccdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_lcdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	alpha
	)

Definition at line 18 of file skew_normal_lcdf.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_lpdf	(	const T_y &	y,
		const T_loc &	mu,
		const T_scale &	sigma,
		const T_shape &	alpha
	)

Definition at line 18 of file skew_normal_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >

return_type_t<T_y, T_loc, T_scale, T_shape> stan::math::skew_normal_lpdf ( const T_y &  y, const T_loc &  mu, const T_scale &  sigma, const T_shape &  alpha  )

inline

Definition at line 114 of file skew_normal_lpdf.hpp.

template<typename T >

Eigen::Matrix<fvar<T>, Eigen::Dynamic, 1> stan::math::softmax ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &  alpha )

inline

Definition at line 12 of file softmax.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::softmax ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v )

inline

Return the softmax of the specified vector.

\( \mbox{softmax}(y) = \frac{\exp(y)} {\sum_{k=1}^K \exp(y_k)}, \)

The entries in the Jacobian of the softmax function are given by \( \begin{array}{l} \displaystyle \frac{\partial}{\partial y_m} \mbox{softmax}(y)[k] \\[8pt] \displaystyle \mbox{ } \ \ \ = \left\{ \begin{array}{ll} \mbox{softmax}(y)[k] \times (1 - \mbox{softmax}(y)[m]) & \mbox{ if } m = k, \mbox{ and} \\[6pt] -\mbox{softmax}(y)[k] \times \mbox{softmax}(y)[m] & \mbox{ if } m \neq k. \end{array} \right. \end{array} \)

Template Parameters

T	type of elements in the vector

Parameters

[in]

v

Vector to transform.

Returns

Unit simplex result of the softmax transform of the vector.

Definition at line 45 of file softmax.hpp.

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::softmax ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > &  alpha )

inline

Return the softmax of the specified Eigen vector.

Softmax is guaranteed to return a simplex.

Parameters

alpha

Unconstrained input vector.

Returns

Softmax of the input.

Exceptions

std::domain_error

If the input vector is size 0.

Definition at line 73 of file softmax.hpp.

template<typename T >

std::vector<T> stan::math::sort_asc ( std::vector< T >  xs )

inline

Return the specified standard vector in ascending order.

Template Parameters

T	Type of elements contained in vector.

Parameters

xs	Vector to order.

Returns

Vector in ascending order.

Exceptions

std::domain_error

If any of the values are NaN.

Definition at line 21 of file sort_asc.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::sort_asc ( Eigen::Matrix< T, R, C >  xs )

inline

Return the specified vector in ascending order.

Template Parameters

T	type of elements in the vector
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

xs	Vector to order.

Returns

Vector in ascending order.

Exceptions

std::domain_error

If any of the values are NaN.

Definition at line 23 of file sort_asc.hpp.

template<typename T >

std::vector<T> stan::math::sort_desc ( std::vector< T >  xs )

inline

Return the specified standard vector in descending order.

Template Parameters

T	Type of elements contained in vector.

Parameters

xs	Vector to order.

Returns

Vector in descending order.

Exceptions

std::domain_error

If any of the values are NaN.

Definition at line 22 of file sort_desc.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::sort_desc ( Eigen::Matrix< T, R, C >  xs )

inline

Return the specified vector in descending order.

Template Parameters

T	type of elements in the vector
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

xs	Vector to order.

Returns

Vector in descending order.

Exceptions

std::domain_error

If any of the values are NaN.

Definition at line 25 of file sort_desc.hpp.

template<bool ascending, typename C >

std::vector<int> stan::math::sort_indices

(

const C &

xs

)

Return an integer array of indices of the specified container sorting the values in ascending or descending order based on the value of the first template prameter.

Template Parameters

ascending	true if sort is in ascending order
C	type of container

Parameters

xs	Container to sort

Returns

sorted version of container

Definition at line 63 of file sort_indices.hpp.

template<typename C >

std::vector<int> stan::math::sort_indices_asc

(

const C &

xs

)

Return a sorted copy of the argument container in ascending order.

Template Parameters

C	type of container

Parameters

xs	Container to sort

Returns

sorted version of container

Definition at line 21 of file sort_indices_asc.hpp.

template<typename C >

std::vector<int> stan::math::sort_indices_desc

(

const C &

xs

)

Return a sorted copy of the argument container in ascending order.

Template Parameters

C	type of container

Parameters

xs	Container to sort

Returns

sorted version of container

Definition at line 21 of file sort_indices_desc.hpp.

template<typename T >

fvar<T> stan::math::sqrt ( const fvar< T > &  x )

inline

Definition at line 13 of file sqrt.hpp.

double stan::math::sqrt ( int  x )

inline

Return the square root of the specified argument.

This version is required to disambiguate sqrt(int).

Parameters

[in]

x

Argument.

Returns

Natural exponential of argument.

Definition at line 17 of file sqrt.hpp.

template<typename T >

apply_scalar_unary<sqrt_fun, T>::return_t stan::math::sqrt ( const T &  x )

inline

Vectorized version of sqrt().

Template Parameters

T	type of container

Parameters

x

container

Returns

Square root of each value in x.

Definition at line 33 of file sqrt.hpp.

var stan::math::sqrt ( const var &  a )

inline

Return the square root of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x}}\).

\[ \mbox{sqrt}(x) = \begin{cases} \textrm{NaN} & x < 0 \\ \sqrt{x} & \mbox{if } x\geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sqrt}(x)}{\partial x} = \begin{cases} \textrm{NaN} & x < 0 \\ \frac{1}{2\sqrt{x}} & x\geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable whose square root is taken.

Returns

Square root of variable.

Definition at line 47 of file sqrt.hpp.

double stan::math::sqrt2 ( )

inline

Return the square root of two.

Returns

Square root of two.

Definition at line 92 of file constants.hpp.

template<typename T >

fvar<T> stan::math::square ( const fvar< T > &  x )

inline

Definition at line 12 of file square.hpp.

double stan::math::square ( double  x )

inline

Return the square of the specified argument.

\(\mbox{square}(x) = x^2\).

The implementation of square(x) is just x * x. Given this, this method is mainly useful in cases where x is not a simple primitive type, particularly when it is an auto-dif type.

Parameters

x	Input to square.

Returns

Square of input.

Definition at line 22 of file square.hpp.

template<typename T >

apply_scalar_unary<square_fun, T>::return_t stan::math::square ( const T &  x )

inline

Vectorized version of square().

Template Parameters

T	type of container

Parameters

x

container

Returns

Each value in x squared.

Definition at line 32 of file square.hpp.

var stan::math::square ( const var &  x )

inline

Return the square of the input variable.

Using square(x) is more efficient than using x * x.

\[ \mbox{square}(x) = \begin{cases} x^2 & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{square}(x)}{\partial x} = \begin{cases} 2x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

x	Variable to square.

Returns

Square of variable.

Definition at line 43 of file square.hpp.

template<typename T1 , typename T2 >

return_type_t<T1, T2> stan::math::squared_distance ( const T1 &  x1, const T2 &  x2  )

inline

Returns the squared distance.

Parameters

x1	First vector.
x2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 21 of file squared_distance.hpp.

template<int R, int C>

double stan::math::squared_distance ( const Eigen::Matrix< double, R, C > &  v1, const Eigen::Matrix< double, R, C > &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 24 of file squared_distance.hpp.

template<typename T , int R, int C>

fvar<T> stan::math::squared_distance ( const Eigen::Matrix< fvar< T >, R, C > &  v1, const Eigen::Matrix< double, R, C > &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

R	Rows at compile time of vector inputs
C	columns at compile time of vector inputs
T	Child scalar type of fvar vector input

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 26 of file squared_distance.hpp.

var stan::math::squared_distance ( const var &  a, const var &  b  )

inline

Returns the log sum of exponentials.

Definition at line 37 of file squared_distance.hpp.

var stan::math::squared_distance ( const var &  a, double  b  )

inline

Returns the log sum of exponentials.

Definition at line 43 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>

double stan::math::squared_distance ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

R1	number of rows in the first vector, can be Eigen::Dynamic
C1	number of columns in the first vector, can be Eigen::Dynamic
R2	number of rows in the second vector, can be Eigen::Dynamic
C2	number of columns in the second vector, can be Eigen::Dynamic

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 48 of file squared_distance.hpp.

var stan::math::squared_distance ( double  a, const var &  b  )

inline

Returns the log sum of exponentials.

Definition at line 49 of file squared_distance.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::squared_distance ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

R1	Rows at compile time of first vector input
C1	Columns at compile time of first vector input
R2	Rows at compile time of second vector input
C2	Columns at compile time of second vector input
T	Child scalar type of fvar vector input

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 51 of file squared_distance.hpp.

template<typename T , int R, int C>

fvar<T> stan::math::squared_distance ( const Eigen::Matrix< double, R, C > &  v1, const Eigen::Matrix< fvar< T >, R, C > &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

R	Rows at compile time of vector inputs
C	columns at compile time of vector inputs
T	Child scalar type of fvar vector input

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 75 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>

var stan::math::squared_distance ( const Eigen::Matrix< var, R1, C1 > &  v1, const Eigen::Matrix< var, R2, C2 > &  v2  )

inline

Definition at line 91 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>

var stan::math::squared_distance ( const Eigen::Matrix< var, R1, C1 > &  v1, const Eigen::Matrix< double, R2, C2 > &  v2  )

inline

Definition at line 99 of file squared_distance.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::squared_distance ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

R1	Rows at compile time of first vector input
C1	Columns at compile time of first vector input
R2	Rows at compile time of second vector input
C2	Columns at compile time of second vector input
T	Child scalar type of fvar vector input

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 100 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>

var stan::math::squared_distance ( const Eigen::Matrix< double, R1, C1 > &  v1, const Eigen::Matrix< var, R2, C2 > &  v2  )

inline

Definition at line 107 of file squared_distance.hpp.

template<typename T , int R, int C>

fvar<T> stan::math::squared_distance ( const Eigen::Matrix< fvar< T >, R, C > &  v1, const Eigen::Matrix< fvar< T >, R, C > &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

R	Rows at compile time of vector inputs
C	columns at compile time of vector inputs
T	Child scalar type of fvar vector input

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 123 of file squared_distance.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::math::squared_distance ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1, const Eigen::Matrix< fvar< T >, R2, C2 > &  v2  )

inline

Returns the squared distance between the specified vectors of the same dimensions.

Template Parameters

R1	Rows at compile time of first vector input
C1	Columns at compile time of first vector input
R2	Rows at compile time of second vector input
C2	Columns at compile time of second vector input
T	Child scalar type of fvar vector input

Parameters

v1	First vector.
v2	Second vector.

Returns

Dot product of the vectors.

Exceptions

std::domain_error

If the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 148 of file squared_distance.hpp.

void stan::math::stan_print ( std::ostream *  o, const var &  x  )

inline

Definition at line 11 of file stan_print.hpp.

template<typename T >

void stan::math::stan_print	(	std::ostream *	o,
		const T &	x
	)

Definition at line 12 of file stan_print.hpp.

template<typename T >

void stan::math::stan_print	(	std::ostream *	o,
		const std::vector< T > &	x
	)

Definition at line 17 of file stan_print.hpp.

template<typename T >

void stan::math::stan_print	(	std::ostream *	o,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x
	)

Definition at line 29 of file stan_print.hpp.

template<typename T >

void stan::math::stan_print	(	std::ostream *	o,
		const Eigen::Matrix< T, 1, Eigen::Dynamic > &	x
	)

Definition at line 41 of file stan_print.hpp.

template<typename T >

void stan::math::stan_print	(	std::ostream *	o,
		const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	x
	)

Definition at line 53 of file stan_print.hpp.

static void stan::math::start_nested ( )

inlinestatic

Record the current position so that recover_memory_nested() can find it.

Definition at line 13 of file start_nested.hpp.

template<typename T_y >

return_type_t<T_y> stan::math::std_normal_lpdf ( const T_y &  y )

inline

Definition at line 58 of file std_normal_lpdf.hpp.

var stan::math::step ( const var &  a )

inline

Return the step, or heaviside, function applied to the specified variable (stan).

See step() for the double-based version.

The derivative of the step function is zero everywhere but at 0, so for convenience, it is taken to be everywhere zero,

\(\mbox{step}(x) = 0\).

Parameters

a	Variable argument.

Returns

The constant variable with value 1.0 if the argument's value is greater than or equal to 0.0, and value 0.0 otherwise.

Definition at line 26 of file step.hpp.

template<typename T >

double stan::math::step ( const T &  y )

inline

The step, or Heaviside, function.

The function is defined by

step(y) = (y < 0.0) ? 0 : 1.

\[ \mbox{step}(x) = \begin{cases} 0 & \mbox{if } x \leq 0 \\ 1 & \mbox{if } x > 0 \\[6pt] 0 & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters

T	type of value

Parameters

y

value

Returns

zero if the value is less than zero, and one otherwise

Definition at line 29 of file step.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::student_t_cdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 19 of file student_t_cdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::student_t_lccdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 19 of file student_t_lccdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::student_t_lcdf	(	const T_y &	y,
		const T_dof &	nu,
		const T_loc &	mu,
		const T_scale &	sigma
	)

Definition at line 19 of file student_t_lcdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >

return_type_t<T_y, T_dof, T_loc, T_scale> stan::math::student_t_lpdf ( const T_y &  y, const T_dof &  nu, const T_loc &  mu, const T_scale &  sigma  )

inline

Definition at line 196 of file student_t_lpdf.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::sub_col ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m, size_t  i, size_t  j, size_t  nrows  )

inline

Return a nrows x 1 subcolumn starting at (i-1, j-1).

Template Parameters

T	type of elements in the matrix

Parameters

m	Matrix.
i	Starting row + 1.
j	Starting column + 1.
nrows	Number of rows in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 21 of file sub_col.hpp.

template<typename T >

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::sub_row ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m, size_t  i, size_t  j, size_t  ncols  )

inline

Return a 1 x nrows subrow starting at (i-1, j-1).

Template Parameters

T	type of elements in the matrix

Parameters

m	Matrix Input matrix.
i	Starting row + 1.
j	Starting column + 1.
ncols	Number of columns in block.

Exceptions

std::out_of_range

if either index is out of range.

Definition at line 21 of file sub_row.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::subtract ( const Eigen::Matrix< T1, R, C > &  m1, const Eigen::Matrix< T2, R, C > &  m2  )

inline

Return the result of subtracting the second specified matrix from the first specified matrix.

The return scalar type is the promotion of the input types.

Template Parameters

T1	type of elements in the first matrix
T2	type of elements in the second matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

m1	First matrix.
m2	Second matrix.

Returns

Difference between first matrix and second matrix.

Definition at line 25 of file subtract.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::subtract ( const T1 &  c, const Eigen::Matrix< T2, R, C > &  m  )

inline

Definition at line 32 of file subtract.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<return_type_t<T1, T2>, R, C> stan::math::subtract ( const Eigen::Matrix< T1, R, C > &  m, const T2 &  c  )

inline

Definition at line 38 of file subtract.hpp.

double stan::math::sum ( double  v )

inline

Returns specified input value.

Template Parameters

T	Type of element.

Parameters

v	Specified value.

Returns

Same value (the sum of one value).

Definition at line 15 of file sum.hpp.

template<typename Derived >

Eigen::DenseBase<Derived>::Scalar stan::math::sum ( const Eigen::DenseBase< Derived > &  v )

inline

Returns the sum of the coefficients of the specified Eigen Matrix, Array or expression.

Template Parameters

Derived

type of argument

Parameters

v

argument

Returns

Sum of coefficients of argument.

Definition at line 19 of file sum.hpp.

template<typename T >

fvar<T> stan::math::sum ( const std::vector< fvar< T > > &  m )

inline

Return the sum of the entries of the specified standard vector.

Template Parameters

T	Type of vector entries.

Parameters

m

Vector.

Returns

Sum of vector entries.

Definition at line 20 of file sum.hpp.

template<typename T >

T stan::math::sum ( const std::vector< T > &  xs )

inline

Return the sum of the values in the specified standard vector.

Template Parameters

T	Type of elements summed.

Parameters

xs	Standard vector to sum.

Returns

Sum of elements.

Definition at line 20 of file sum.hpp.

template<typename T , int R, int C>

fvar<T> stan::math::sum ( const Eigen::Matrix< fvar< T >, R, C > &  m )

inline

Return the sum of the entries of the specified matrix.

Template Parameters

T	Type of matrix entries.
R	Row type of matrix.
C	Column type of matrix.

Parameters

m

Matrix.

Returns

Sum of matrix entries.

Definition at line 21 of file sum.hpp.

int stan::math::sum ( int  v )

inline

Returns specified input value.

Template Parameters

T	Type of element.

Parameters

v	Specified value.

Returns

Same value (the sum of one value).

Definition at line 24 of file sum.hpp.

template<int R, int C>

var stan::math::sum ( const Eigen::Matrix< var, R, C > &  m )

inline

Returns the sum of the coefficients of the specified matrix, column vector or row vector.

Template Parameters

R	Row type for matrix.
C	Column type for matrix.

Parameters

m	Specified matrix or vector.

Returns

Sum of coefficients of matrix.

Definition at line 46 of file sum.hpp.

var stan::math::sum ( const std::vector< var > &  m )

inline

Returns the sum of the entries of the specified vector.

Parameters

m

Vector.

Returns

Sum of vector entries.

Definition at line 54 of file sum.hpp.

void stan::math::system_error ( const char *  function, const char *  name, const int &  y, const char *  msg1, const char *  msg2  )

inline

Throw a system error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing system errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name> <msg1><y><msg2>"

Parameters

[in]	function	Name of the function.
[in]	name	Name of the variable.
[in]	y	Error code.
[in]	msg1	Message to print before the variable.
[in]	msg2	Message to print after the variable.

Exceptions

std::system_error

Always.

Definition at line 25 of file system_error.hpp.

void stan::math::system_error ( const char *  function, const char *  name, const int &  y, const char *  msg1  )

inline

Throw a system error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing system errors. This will allow us to change the behavior for all functions at once. The message is: * "<function>: <name> <msg1><y>"

Parameters

[in]	function	Name of the function.
[in]	name	Name of the variable.
[in]	y	Error code.
[in]	msg1	Message to print before the variable.

Exceptions

std::system_error

Always.

Definition at line 45 of file system_error.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::tail ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v, size_t  n  )

inline

Return the specified number of elements as a vector from the back of the specified vector.

Template Parameters

T	type of elements in the vector

Parameters

v	Vector input.
n	Size of return.

Returns

The last n elements of v.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 23 of file tail.hpp.

template<typename T >

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::tail ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  rv, size_t  n  )

inline

Return the specified number of elements as a row vector from the back of the specified row vector.

Template Parameters

T	type of elements in the vector

Parameters

rv	Row vector.
n	Size of return row vector.

Returns

The last n elements of rv.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 42 of file tail.hpp.

template<typename T >

std::vector<T> stan::math::tail	(	const std::vector< T > &	sv,
		size_t	n
	)

Return the specified number of elements as a standard vector from the back of the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

sv	Standard vector.
n	Size of return.

Returns

The last n elements of sv.

Exceptions

std::out_of_range

if n is out of range.

Definition at line 61 of file tail.hpp.

template<typename T >

fvar<T> stan::math::tan ( const fvar< T > &  x )

inline

Definition at line 11 of file tan.hpp.

template<typename T >

apply_scalar_unary<tan_fun, T>::return_t stan::math::tan ( const T &  x )

inline

Vectorized version of tan().

Template Parameters

T	type of container

Parameters

x	angles in radians

Returns

Tangent of each value in x.

Definition at line 33 of file tan.hpp.

var stan::math::tan ( const var &  a )

inline

Return the tangent of a radian-scaled variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \tan x = \sec^2 x\).

\[ \mbox{tan}(x) = \begin{cases} \tan(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tan}(x)}{\partial x} = \begin{cases} \sec^2(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Variable for radians of angle.

Returns

Tangent of variable.

Definition at line 46 of file tan.hpp.

template<typename T >

fvar<T> stan::math::tanh ( const fvar< T > &  x )

inline

Definition at line 12 of file tanh.hpp.

template<typename T >

apply_scalar_unary<tanh_fun, T>::return_t stan::math::tanh ( const T &  x )

inline

Vectorized version of tanh().

Template Parameters

T	type of container

Parameters

x	angles in radians

Returns

Hyperbolic tangent of each value in x.

Definition at line 33 of file tanh.hpp.

var stan::math::tanh ( const var &  a )

inline

Return the hyperbolic tangent of the specified variable (cmath).

The derivative is defined by

\(\frac{d}{dx} \tanh x = \frac{1}{\cosh^2 x}\).

\[ \mbox{tanh}(x) = \begin{cases} \tanh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tanh}(x)}{\partial x} = \begin{cases} \mbox{sech}^2(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a

Variable.

Returns

Hyperbolic tangent of variable.

Definition at line 49 of file tanh.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, R> stan::math::tcrossprod ( const Eigen::Matrix< fvar< T >, R, C > &  m )

inline

Definition at line 12 of file tcrossprod.hpp.

template<int R, int C>

Eigen::MatrixXd stan::math::tcrossprod ( const Eigen::Matrix< double, R, C > &  M )

inline

Returns the result of post-multiplying a matrix by its own transpose.

Template Parameters

R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

M	Matrix to multiply.

Returns

M times its transpose.

Definition at line 20 of file tcrossprod.hpp.

template<int R, int C>

Eigen::Matrix<var, -1, -1> stan::math::tcrossprod ( const Eigen::Matrix< var, R, C > &  M )

inline

Returns the result of post-multiplying a matrix by its own transpose.

Parameters

M	Matrix to multiply.

Returns

M times its transpose.

Definition at line 22 of file tcrossprod.hpp.

double stan::math::tgamma ( double  x )

inline

Return the gamma function applied to the specified argument.

Parameters

x

Argument.

Returns

The gamma function applied to argument.

Definition at line 18 of file tgamma.hpp.

template<typename T >

fvar<T> stan::math::tgamma ( const fvar< T > &  x )

inline

Return the result of applying the gamma function to the specified argument.

Template Parameters

T	Scalar type of autodiff variable.

Parameters

x

Argument.

Returns

Gamma function applied to argument.

Definition at line 21 of file tgamma.hpp.

template<typename T >

apply_scalar_unary<tgamma_fun, T>::return_t stan::math::tgamma ( const T &  x )

inline

Vectorized version of tgamma().

Template Parameters

T	type of container

Parameters

x

container

Returns

Gamma function applied to each value in x.

Exceptions

std::domain_error

if any value is 0 or a negative integer

Definition at line 34 of file tgamma.hpp.

var stan::math::tgamma ( const var &  a )

inline

Return the Gamma function applied to the specified variable (C99).

The derivative with respect to the argument is

\(\frac{d}{dx} \Gamma(x) = \Gamma(x) \Psi^{(0)}(x)\)

where \(\Psi^{(0)}(x)\) is the digamma function.

\[ \mbox{tgamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Gamma(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tgamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Gamma(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Gamma(x)=\int_0^{\infty} u^{x - 1} \exp(-u) \, du \]

\[ \frac{\partial \, \Gamma(x)}{\partial x} = \Gamma(x)\Psi(x) \]

Parameters

a	Argument to function.

Returns

The Gamma function applied to the specified argument.

Definition at line 56 of file tgamma.hpp.

template<typename T >

void stan::math::throw_domain_error ( const char *  function, const char *  name, const T &  y, const char *  msg1, const char *  msg2  )

inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name> <msg1><y><msg2>"

Template Parameters

T	Type of variable.

Parameters

[in]	function	Name of the function.
[in]	name	Name of the variable.
[in]	y	Variable.
[in]	msg1	Message to print before the variable.
[in]	msg2	Message to print after the variable.

Exceptions

std::domain_error

Always.

Definition at line 26 of file throw_domain_error.hpp.

template<typename T >

void stan::math::throw_domain_error ( const char *  function, const char *  name, const T &  y, const char *  msg1  )

inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: * "<function>: <name> <msg1><y>"

Template Parameters

T	Type of variable.

Parameters

[in]	function	Name of the function.
[in]	name	Name of the variable.
[in]	y	Variable.
[in]	msg1	Message to print before the variable.

Exceptions

std::domain_error

Always.

Definition at line 49 of file throw_domain_error.hpp.

template<typename T >

void stan::math::throw_domain_error_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg1, const char *  msg2  )

inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Index
msg1	Message to print before the variable
msg2	Message to print after the variable

Exceptions

std::domain_error

Always.

Definition at line 30 of file throw_domain_error_vec.hpp.

template<typename T >

void stan::math::throw_domain_error_vec ( const char *  function, const char *  name, const T &  y, size_t  i, const char *  msg  )

inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters

T	Type of variable

Parameters

function	Name of the function
name	Name of the variable
y	Variable
i	Index
msg	Message to print before the variable

Exceptions

std::domain_error

Always

Definition at line 56 of file throw_domain_error_vec.hpp.

template<typename T , int R, int C>

std::vector<T> stan::math::to_array_1d ( const Eigen::Matrix< T, R, C > &  matrix )

inline

Definition at line 15 of file to_array_1d.hpp.

template<typename T >

std::vector<T> stan::math::to_array_1d ( const std::vector< T > &  x )

inline

Definition at line 27 of file to_array_1d.hpp.

template<typename T >

std::vector<typename scalar_type<T>::type> stan::math::to_array_1d ( const std::vector< std::vector< T > > &  x )

inline

Definition at line 33 of file to_array_1d.hpp.

template<typename T >

std::vector<std::vector<T> > stan::math::to_array_2d ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  matrix )

inline

Definition at line 12 of file to_array_2d.hpp.

template<typename T >

fvar<T> stan::math::to_fvar ( const T &  x )

inline

Definition at line 11 of file to_fvar.hpp.

template<typename T >

std::vector<fvar<T> > stan::math::to_fvar ( const std::vector< T > &  v )

inline

Definition at line 12 of file to_fvar.hpp.

template<int R, int C, typename T >

const Eigen::Matrix<T, R, C>& stan::math::to_fvar ( const Eigen::Matrix< T, R, C > &  m )

inline

Specialization of to_fvar for const matrices of fvars.

Parameters

[in,out]

m

A matrix of forward automatic differentation variables.

Returns

The input matrix of forward automatic differentiation variables.

Definition at line 20 of file to_fvar.hpp.

template<typename T >

std::vector<fvar<T> > stan::math::to_fvar ( const std::vector< T > &  v, const std::vector< T > &  d  )

inline

Definition at line 21 of file to_fvar.hpp.

template<typename T >

const fvar<T>& stan::math::to_fvar ( const fvar< T > &  x )

inline

Specialization of to_fvar for const fvars.

Parameters

[in,out]

x

A forward automatic differentation variables.

Returns

The input forward automatic differentiation variables.

Definition at line 23 of file to_fvar.hpp.

template<int R, int C, typename T >

Eigen::Matrix<T, R, C>& stan::math::to_fvar ( Eigen::Matrix< T, R, C > &  m )

inline

Specialization of to_fvar for non-const matrices of fvars.

Parameters

[in,out]

m

A matrix of forward automatic differentation variables.

Returns

The input matrix of forward automatic differentiation variables.

Definition at line 32 of file to_fvar.hpp.

template<typename T >

fvar<T>& stan::math::to_fvar ( fvar< T > &  x )

inline

Specialization of to_fvar for non-const fvars.

Parameters

[in,out]

x

A forward automatic differentation variables.

Returns

The input forward automatic differentiation variables.

Definition at line 35 of file to_fvar.hpp.

template<int R, int C>

Eigen::Matrix<fvar<double>, R, C> stan::math::to_fvar ( const Eigen::Matrix< double, R, C > &  m )

inline

Definition at line 37 of file to_fvar.hpp.

template<typename T >

const std::vector<fvar<T> >& stan::math::to_fvar ( const std::vector< fvar< T >> &  v )

inline

Specialization of to_fvar for const fvar input.

Template Parameters

The	inner type of the fvar.

Parameters

[in,out]

v

A vector of forward automatic differentiation variable.

Returns

The input vector of forward automatic differentiation variable.

Definition at line 38 of file to_fvar.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::to_fvar ( const Eigen::Matrix< T, R, C > &  val, const Eigen::Matrix< T, R, C > &  deriv  )

inline

Definition at line 47 of file to_fvar.hpp.

template<typename T >

std::vector<fvar<T> >& stan::math::to_fvar ( std::vector< fvar< T >> &  v )

inline

Specialization of to_fvar for non-const fvar input.

Template Parameters

The	inner type of the fvar.

Parameters

[in,out]

v

A vector of forward automatic differentiation variable.

Returns

The input vector of forward automatic differentiation variable.

Definition at line 50 of file to_fvar.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const Eigen::Matrix< T, R, C > &  x )

inline

Returns a matrix with dynamic dimensions constructed from an Eigen matrix which is either a row vector, column vector, or matrix.

The runtime dimensions will be the same as the input.

Template Parameters

T	type of the elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

x

matrix

Returns

the matrix representation of the input

Definition at line 25 of file to_matrix.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< Eigen::Matrix< T, 1, Eigen::Dynamic > > &  x )

inline

Returns a matrix representation of a standard vector of Eigen row vectors with the same dimensions and indexing order.

Template Parameters

T	type of the elements in the vector

Parameters

x	Eigen vector of vectors of scalar values

Returns

the matrix representation of the input

Definition at line 39 of file to_matrix.hpp.

template<typename T >

Eigen::Matrix<return_type_t<T, double>, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< std::vector< T > > &  x )

inline

Returns a matrix representation of the standard vector of standard vectors with the same dimensions and indexing order.

Template Parameters

T	type of elements in the vector

Parameters

x	vector of vectors of scalar values

Returns

the matrix representation of the input

Definition at line 65 of file to_matrix.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const Eigen::Matrix< T, R, C > &  x, int  m, int  n  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

x	matrix
m	rows
n	columns

Returns

Reshaped inputted matrix

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 98 of file to_matrix.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< T > &  x, int  m, int  n  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Template Parameters

T	type of elements in the vector

Parameters

x	vector of values
m	rows
n	columns

Returns

the matrix representation of the input

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 120 of file to_matrix.hpp.

Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< int > &  x, int  m, int  n  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Parameters

x	vector of values
m	rows
n	columns

Returns

the matrix representation of the input

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 139 of file to_matrix.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const Eigen::Matrix< T, R, C > &  x, int  m, int  n, bool  col_major  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Template Parameters

T	type of elements in the matrix
R	number of rows, can be Eigen::Dynamic
C	number of columns, can be Eigen::Dynamic

Parameters

x	matrix
m	rows
n	columns
col_major	column-major indicator: if 1, output matrix is transversed in column-major order, if 0, output matrix is transversed in row-major order, otherwise function throws std::invalid_argument

Returns

Reshaped inputted matrix

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 171 of file to_matrix.hpp.

template<typename T >

Eigen::Matrix<return_type_t<T, double>, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< T > &  x, int  m, int  n, bool  col_major  )

inline

Returns a matrix representation of the vector in column-major order with the specified number of rows and columns.

Template Parameters

T	type of elements in the vector

Parameters

x	vector of values
m	rows
n	columns
col_major	column-major indicator: if 1, output matrix is transversed in column-major order, if 0, output matrix is transversed in row-major order, otherwise function throws std::invalid_argument

Returns

the matrix representation of the input

Exceptions

<code>std::invalid_argument</code>

if the sizes do not match

Definition at line 205 of file to_matrix.hpp.

template<typename T , int R, int C, require_var_t< T > ...>

matrix_cl<T> stan::math::to_matrix_cl ( const Eigen::Matrix< T, R, C > &  src )

inline

Copies the source Eigen matrix of vars to the destination matrix that is stored on the OpenCL device.

Template Parameters

R	Compile time rows of the Eigen matrix
C	Compile time columns of the Eigen matrix

Parameters

src	source Eigen matrix

Returns

matrix_cl with a copy of the data in the source matrix

Definition at line 30 of file copy.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::to_row_vector ( const Eigen::Matrix< T, R, C > &  matrix )

inline

Definition at line 14 of file to_row_vector.hpp.

template<typename T >

Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::to_row_vector ( const std::vector< T > &  vec )

inline

Definition at line 22 of file to_row_vector.hpp.

Eigen::Matrix<double, 1, Eigen::Dynamic> stan::math::to_row_vector ( const std::vector< int > &  vec )

inline

Definition at line 28 of file to_row_vector.hpp.

var stan::math::to_var ( double  x )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

x

A scalar value

Returns

An automatic differentiation variable with the input value.

Definition at line 18 of file to_var.hpp.

std::vector<var> stan::math::to_var ( const std::vector< double > &  v )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

v

A std::vector<double>

Returns

A std::vector<var> with the values set

Definition at line 19 of file to_var.hpp.

matrix_v stan::math::to_var ( const matrix_d &  m )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

m

A Matrix with scalars

Returns

A Matrix with automatic differentiation variables

Definition at line 22 of file to_var.hpp.

var& stan::math::to_var ( var &  x )

inline

Specialization of to_var for non-const var input.

Parameters

[in,out]

x

An automatic differentiation variable.

Returns

The input automatic differentiation variable.

Definition at line 27 of file to_var.hpp.

matrix_v& stan::math::to_var ( matrix_v &  m )

inline

Specialization of to_var for non-const matrices of vars.

Parameters

[in,out]

m

A matrix of automatic differentation variables.

Returns

The input matrix of automatic differentiation variables.

Definition at line 34 of file to_var.hpp.

const std::vector<var>& stan::math::to_var ( const std::vector< var > &  v )

inline

Specialization of to_var to for const input vector of var.

Returns a var variable from the input

Parameters

[in]

v

A std::vector<var>

Returns

The input std::vector<var>

Definition at line 35 of file to_var.hpp.

const var& stan::math::to_var ( const var &  x )

inline

Specialization of to_var for const var input.

Parameters

[in,out]

x

An automatic differentiation variable.

Returns

The input automatic differentiation variable.

Definition at line 36 of file to_var.hpp.

const matrix_v& stan::math::to_var ( const matrix_v &  m )

inline

Specialization of to_var for const matrices of vars.

Parameters

[in,out]

m

A matrix of automatic differentation variables.

Returns

The input matrix of automatic differentiation variables.

Definition at line 43 of file to_var.hpp.

std::vector<var>& stan::math::to_var ( std::vector< var > &  v )

inline

Specialization of to_var to for non-const input vector of var.

Returns a var variable from the input

Parameters

[in]

v

A std::vector<var>

Returns

The input std::vector<var>

Definition at line 45 of file to_var.hpp.

vector_v stan::math::to_var ( const vector_d &  v )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

v

A Vector of scalars

Returns

A Vector of automatic differentiation variables with values of v

Definition at line 54 of file to_var.hpp.

const vector_v& stan::math::to_var ( const vector_v &  v )

inline

Specialization of to_var for const column vector of vars.

Parameters

[in,out]

v

A column vector of automatic differentation variables.

Returns

The input column vector of automatic differentiation variables.

Definition at line 66 of file to_var.hpp.

vector_v& stan::math::to_var ( vector_v &  v )

inline

Specialization of to_var for non-const column vector of vars.

Parameters

[in,out]

v

A column vector of automatic differentation variables.

Returns

The input column vector of automatic differentiation variables.

Definition at line 75 of file to_var.hpp.

row_vector_v stan::math::to_var ( const row_vector_d &  rv )

inline

Converts argument to an automatic differentiation variable.

Returns a var variable with the input value.

Parameters

[in]

rv

A row vector of scalars

Returns

A row vector of automatic differentation variables with values of rv.

Definition at line 86 of file to_var.hpp.

const row_vector_v& stan::math::to_var ( const row_vector_v &  rv )

inline

Specialization of to_var for const row vector of vars.

Parameters

[in,out]

rv

A column vector of automatic differentation variables.

Returns

The input row vector of automatic differentiation variables.

Definition at line 98 of file to_var.hpp.

row_vector_v& stan::math::to_var ( row_vector_v &  rv )

inline

Specialization of to_var for non-const row vector of vars.

Parameters

[in,out]

rv

A column vector of automatic differentation variables.

Returns

The input row vector of automatic differentiation variables.

Definition at line 107 of file to_var.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::to_vector ( const Eigen::Matrix< T, R, C > &  matrix )

inline

Definition at line 14 of file to_vector.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::to_vector ( const std::vector< T > &  vec )

inline

Definition at line 22 of file to_vector.hpp.

Eigen::Matrix<double, Eigen::Dynamic, 1> stan::math::to_vector ( const std::vector< int > &  vec )

inline

Definition at line 28 of file to_vector.hpp.

template<typename T >

T stan::math::trace ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m )

inline

Returns the trace of the specified matrix.

The trace is defined as the sum of the elements on the diagonal. The matrix is not required to be square. Returns 0 if matrix is empty.

Template Parameters

T	type of the elements in the matrix

Parameters

[in]

m

Specified matrix.

Returns

Trace of the matrix.

Definition at line 20 of file trace.hpp.

template<typename T >

T stan::math::trace ( const T &  m )

inline

Definition at line 25 of file trace.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename T3 , int R3, int C3, require_any_var_t< T1, T2, T3 > ...>

var stan::math::trace_gen_inv_quad_form_ldlt ( const Eigen::Matrix< T1, R1, C1 > &  D, const LDLT_factor< T2, R2, C2 > &  A, const Eigen::Matrix< T3, R3, C3 > &  B  )

inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(D B^T A^-1 B) where D is a square matrix and the LDLT_factor of A is provided.

Definition at line 21 of file trace_gen_inv_quad_form_ldlt.hpp.

template<typename T1 , typename T2 , typename T3 , int R1, int C1, int R2, int C2, int R3, int C3, typename = require_any_not_var_t<T1, T2, T3>>

return_type_t<T1, T2, T3> stan::math::trace_gen_inv_quad_form_ldlt ( const Eigen::Matrix< T1, R1, C1 > &  D, const LDLT_factor< T2, R2, C2 > &  A, const Eigen::Matrix< T3, R3, C3 > &  B  )

inline

Definition at line 41 of file trace_gen_inv_quad_form_ldlt.hpp.

template<typename TD , int RD, int CD, typename TA , int RA, int CA, typename TB , int RB, int CB, typename = require_any_not_var_t<TD, TA, TB>>

return_type_t<TD, TA, TB> stan::math::trace_gen_quad_form ( const Eigen::Matrix< TD, RD, CD > &  D, const Eigen::Matrix< TA, RA, CA > &  A, const Eigen::Matrix< TB, RB, CB > &  B  )

inline

Return the trace of D times the quadratic form of B and A.

That is, `trace_gen_quad_form(D, A, B) = trace(D * B' * A * B).`

Template Parameters

TD	type of elements in the first matrix
TA	type of elements in the second matrix
TB	type of elements in the third matrix
RD	number of rows in the first matrix, can be Eigen::Dynamic
CD	number of columns in the first matrix, can be Eigen::Dynamic
RA	number of rows in the second matrix, can be Eigen::Dynamic
CA	number of columns in the second matrix, can be Eigen::Dynamic
RB	number of rows in the third matrix, can be Eigen::Dynamic
CB	number of columns in the third matrix, can be Eigen::Dynamic

Parameters

D	multiplier
A	outside term in quadratic form
B	inner term in quadratic form

Returns

trace(D * B' * A * B)

Exceptions

std::domain_error	if A or D is not square
std::domain_error	if A cannot be multiplied by B or B cannot be multiplied by D.

Definition at line 39 of file trace_gen_quad_form.hpp.

template<int RD, int CD, int RA, int CA, typename TB , int RB, int CB>

double stan::math::trace_gen_quad_form ( const Eigen::Matrix< double, RD, CD > &  D, const Eigen::Matrix< double, RA, CA > &  A, const Eigen::Matrix< double, RB, CB > &  B  )

inline

Return the trace of D times the quadratic form of B and A.

That is, `trace_gen_quad_form(D, A, B) = trace(D * B' * A * B).` This is the double-only overload to allow Eigen's expression templates to be used for efficiency.

Template Parameters

RD	number of rows in the first matrix, can be Eigen::Dynamic
CD	number of columns in the first matrix, can be Eigen::Dynamic
RA	number of rows in the second matrix, can be Eigen::Dynamic
CA	number of columns in the second matrix, can be Eigen::Dynamic
TB	type of elements in the third matrix
RB	number of rows in the third matrix, can be Eigen::Dynamic
CB	number of columns in the third matrix, can be Eigen::Dynamic

Parameters

D	multiplier
A	outside term in quadratic form
B	inner term in quadratic form

Returns

trace(D * B' * A * B)

Exceptions

std::domain_error	if A or D is not square
std::domain_error	if A cannot be multiplied by B or B cannot be multiplied by D.

Definition at line 72 of file trace_gen_quad_form.hpp.

template<typename Td , int Rd, int Cd, typename Ta , int Ra, int Ca, typename Tb , int Rb, int Cb, typename = require_any_var_t<Td, Ta, Tb>>

var stan::math::trace_gen_quad_form ( const Eigen::Matrix< Td, Rd, Cd > &  D, const Eigen::Matrix< Ta, Ra, Ca > &  A, const Eigen::Matrix< Tb, Rb, Cb > &  B  )

inline

Definition at line 89 of file trace_gen_quad_form.hpp.

template<typename T1 , typename T2 , int R2, int C2, int R3, int C3, typename = require_any_not_var_t<T1, T2>>

return_type_t<T1, T2> stan::math::trace_inv_quad_form_ldlt ( const LDLT_factor< T1, R2, C2 > &  A, const Eigen::Matrix< T2, R3, C3 > &  B  )

inline

Definition at line 33 of file trace_inv_quad_form_ldlt.hpp.

template<typename T2 , int R2, int C2, typename T3 , int R3, int C3, typename = require_any_var_t<T2, T3>>

return_type_t<T2, T3> stan::math::trace_inv_quad_form_ldlt ( const LDLT_factor< T2, R2, C2 > &  A, const Eigen::Matrix< T3, R3, C3 > &  B  )

inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(B^T A^-1 B) where the LDLT_factor of A is provided.

Definition at line 148 of file trace_inv_quad_form_ldlt.hpp.

template<int RA, int CA, int RB, int CB, typename T >

fvar<T> stan::math::trace_quad_form ( const Eigen::Matrix< fvar< T >, RA, CA > &  A, const Eigen::Matrix< fvar< T >, RB, CB > &  B  )

inline

Definition at line 15 of file trace_quad_form.hpp.

template<int RA, int CA, int RB, int CB, typename T >

fvar<T> stan::math::trace_quad_form ( const Eigen::Matrix< fvar< T >, RA, CA > &  A, const Eigen::Matrix< double, RB, CB > &  B  )

inline

Definition at line 23 of file trace_quad_form.hpp.

template<int RA, int CA, int RB, int CB>

double stan::math::trace_quad_form ( const Eigen::Matrix< double, RA, CA > &  A, const Eigen::Matrix< double, RB, CB > &  B  )

inline

Compute trace(B^T A B).

Template Parameters

RA	number of rows in the first matrix, can be Eigen::Dynamic
CA	number of columns in the first matrix, can be Eigen::Dynamic
RB	number of rows in the second matrix, can be Eigen::Dynamic
CB	number of columns in the second matrix, can be Eigen::Dynamic

Parameters

A	matrix
B	matrix

Returns

The trace of B^T A B

Exceptions

std::domain_error	if A is not square
std::domain_error	if A cannot be multiplied by B

Definition at line 25 of file trace_quad_form.hpp.

template<int RA, int CA, int RB, int CB, typename T >

fvar<T> stan::math::trace_quad_form ( const Eigen::Matrix< double, RA, CA > &  A, const Eigen::Matrix< fvar< T >, RB, CB > &  B  )

inline

Definition at line 31 of file trace_quad_form.hpp.

template<typename Ta , int Ra, int Ca, typename Tb , int Rb, int Cb, typename = require_any_var_t<Ta, Tb>>

return_type_t<Ta, Tb> stan::math::trace_quad_form ( const Eigen::Matrix< Ta, Ra, Ca > &  A, const Eigen::Matrix< Tb, Rb, Cb > &  B  )

inline

Definition at line 78 of file trace_quad_form.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, C, R> stan::math::transpose ( const Eigen::Matrix< T, R, C > &  m )

inline

Definition at line 10 of file transpose.hpp.

template<typename T >

fvar<T> stan::math::trigamma ( const fvar< T > &  u )

inline

Return the value of the trigamma function at the specified argument (i.e., the second derivative of the log Gamma function at the specified argument).

Parameters

u

argument

Returns

trigamma function at argument

Definition at line 20 of file trigamma.hpp.

var stan::math::trigamma ( const var &  u )

inline

Return the value of the trigamma function at the specified argument (i.e., the second derivative of the log Gamma function at the specified argument).

Parameters

u

argument

Returns

trigamma function at argument

Definition at line 23 of file trigamma.hpp.

template<typename T >

apply_scalar_unary<trigamma_fun, T>::return_t stan::math::trigamma ( const T &  x )

inline

Return the elementwise application of trigamma() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise trigamma of container elements

Definition at line 39 of file trigamma.hpp.

double stan::math::trigamma ( double  u )

inline

Return the second derivative of the log Gamma function evaluated at the specified argument.

\[ \mbox{trigamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi_1(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{trigamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Psi_1(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Psi_1(x)=\sum_{n=0}^\infty \frac{1}{(x+n)^2} \]

\[ \frac{\partial \, \Psi_1(x)}{\partial x} = -2\sum_{n=0}^\infty \frac{1}{(x+n)^3} \]

Parameters

[in]

u

argument

Returns

second derivative of log Gamma function at argument

Definition at line 116 of file trigamma.hpp.

double stan::math::trigamma ( int  u )

inline

Return the second derivative of the log Gamma function evaluated at the specified argument.

Parameters

[in]

u

argument

Returns

second derivative of log Gamma function at argument

Definition at line 125 of file trigamma.hpp.

template<typename T >

T stan::math::trigamma_impl ( const T &  x )

inline

Return the trigamma function applied to the argument.

The trigamma function returns the second derivative of the log Gamma function evaluated at the specified argument. This base templated version is used in prim, fwd, and rev implementations.

Template Parameters

T	scalar argument type

Parameters

x

argument

Returns

second derivative of log Gamma function at argument

Definition at line 33 of file trigamma.hpp.

double stan::math::trunc ( double  x )

inline

Return the nearest integral value that is not larger in magnitude than the specified argument.

Parameters

[in]

x

Argument.

Returns

The truncated argument.

Definition at line 17 of file trunc.hpp.

template<typename T >

fvar<T> stan::math::trunc ( const fvar< T > &  x )

inline

Return the nearest integral value that is not larger in magnitude than the specified argument.

Template Parameters

T	Scalar type of autodiff variable.

Parameters

[in]

x

Argument.

Returns

The truncated argument.

Definition at line 20 of file trunc.hpp.

double stan::math::trunc ( int  x )

inline

Return the nearest integral value that is not larger in magnitude than the specified argument.

Parameters

[in]

x

Argument.

Returns

The truncated argument.

Definition at line 26 of file trunc.hpp.

template<typename T >

apply_scalar_unary<trunc_fun, T>::return_t stan::math::trunc ( const T &  x )

inline

Return the elementwise application of trunc() to specified argument container.

The return type promotes the underlying scalar argument type to double if it is an integer, and otherwise is the argument type.

Template Parameters

T	type of container

Parameters

x

container

Returns

elementwise trunc of container elements

Definition at line 39 of file trunc.hpp.

var stan::math::trunc ( const var &  a )

inline

Returns the truncatation of the specified variable (C99).

See trunc() for the double-based version.

The derivative is zero everywhere but at integer values, so for convenience the derivative is defined to be everywhere zero,

\(\frac{d}{dx} \mbox{trunc}(x) = 0\).

\[ \mbox{trunc}(x) = \begin{cases} \lfloor x \rfloor & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{trunc}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters

a	Specified variable.

Returns

Truncation of the variable.

Definition at line 55 of file trunc.hpp.

template<typename T , typename U >

return_type_t<T, U> stan::math::ub_constrain ( const T &  x, const U &  ub  )

inline

Return the upper-bounded value for the specified unconstrained scalar and upper bound.

The transform is

\(f(x) = U - \exp(x)\)

where \(U\) is the upper bound.

If the upper bound is positive infinity, this function reduces to identity_constrain(x).

Template Parameters

T	type of scalar
U	type of upper bound

Parameters

[in]	x	free scalar.
[in]	ub	upper bound

Returns

scalar constrained to have upper bound

Definition at line 32 of file ub_constrain.hpp.

template<typename T , typename U >

return_type_t<T, U> stan::math::ub_constrain ( const T &  x, const U &  ub, T &  lp  )

inline

Return the upper-bounded value for the specified unconstrained scalar and upper bound and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The transform is as specified for ub_constrain(T, double). The log absolute Jacobian determinant is

\( \log | \frac{d}{dx} -\mbox{exp}(x) + U | = \log | -\mbox{exp}(x) + 0 | = x\).

If the upper bound is positive infinity, this function reduces to identity_constrain(x, lp).

Template Parameters

T	type of scalar
U	type of upper bound

Parameters

[in]	x	free scalar.
[in]	ub	upper bound
[in,out]	lp	log density

Returns

scalar constrained to have upper bound

Definition at line 64 of file ub_constrain.hpp.

template<typename T , typename U >

return_type_t<T, U> stan::math::ub_free ( const T &  y, const U &  ub  )

inline

Return the free scalar that corresponds to the specified upper-bounded value with respect to the specified upper bound.

The transform is the reverse of the ub_constrain(T, double) transform,

\(f^{-1}(y) = \log -(y - U)\)

where \(U\) is the upper bound.

If the upper bound is positive infinity, this function reduces to identity_free(y).

Template Parameters

T	type of scalar
U	type of upper bound

Parameters

y	constrained scalar with specified upper bound
ub	upper bound

Returns

unconstrained scalar with respect to upper bound

Exceptions

std::invalid_argument

if constrained scalar is greater than the upper bound.

Definition at line 36 of file ub_free.hpp.

template<typename T_y , typename T_low , typename T_high >

return_type_t<T_y, T_low, T_high> stan::math::uniform_cdf	(	const T_y &	y,
		const T_low &	alpha,
		const T_high &	beta
	)

Definition at line 13 of file uniform_cdf.hpp.

template<typename T_y , typename T_low , typename T_high >

return_type_t<T_y, T_low, T_high> stan::math::uniform_lccdf	(	const T_y &	y,
		const T_low &	alpha,
		const T_high &	beta
	)

Definition at line 15 of file uniform_lccdf.hpp.

template<typename T_y , typename T_low , typename T_high >

return_type_t<T_y, T_low, T_high> stan::math::uniform_lcdf	(	const T_y &	y,
		const T_low &	alpha,
		const T_high &	beta
	)

Definition at line 15 of file uniform_lcdf.hpp.

template<typename T_y , typename T_low , typename T_high >

return_type_t<T_y, T_low, T_high> stan::math::uniform_lpdf ( const T_y &  y, const T_low &  alpha, const T_high &  beta  )

inline

Definition at line 110 of file uniform_lpdf.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::unit_vector_constrain ( const Eigen::Matrix< fvar< T >, R, C > &  y )

inline

Definition at line 20 of file unit_vector_constrain.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::unit_vector_constrain

(

const Eigen::Matrix< T, R, C > &

y

)

Return the unit length vector corresponding to the free vector y.

See the Wikipedia page on generating random points on an N-sphere.

Template Parameters

T	type of elements in the vector

Parameters

y	vector of K unrestricted variables

Returns

Unit length vector of dimension K

Definition at line 24 of file unit_vector_constrain.hpp.

template<typename T , int R, int C>

Eigen::Matrix<T, R, C> stan::math::unit_vector_constrain	(	const Eigen::Matrix< T, R, C > &	y,
		T &	lp
	)

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Template Parameters

T	type of elements in the vector
R	number of rows in the matrix, can be Eigen::Dynamic
C	number of columns in the matrix, can be Eigen::Dynamic

Parameters

y	vector of K unrestricted variables

Returns

Unit length vector of dimension K

Parameters

lp	Log probability reference to increment.

Definition at line 46 of file unit_vector_constrain.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::math::unit_vector_constrain ( const Eigen::Matrix< fvar< T >, R, C > &  y, fvar< T > &  lp  )

inline

Definition at line 52 of file unit_vector_constrain.hpp.

template<int R, int C>

Eigen::Matrix<var, R, C> stan::math::unit_vector_constrain

(

const Eigen::Matrix< var, R, C > &

y

)

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Parameters

y	vector of K unrestricted variables

Returns

Unit length vector of dimension K

Template Parameters

T	Scalar type.

Definition at line 55 of file unit_vector_constrain.hpp.

template<int R, int C>

Eigen::Matrix<var, R, C> stan::math::unit_vector_constrain	(	const Eigen::Matrix< var, R, C > &	y,
		var &	lp
	)

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Parameters

y	vector of K unrestricted variables

Returns

Unit length vector of dimension K

Parameters

lp	Log probability reference to increment.

Template Parameters

T	Scalar type.

Definition at line 90 of file unit_vector_constrain.hpp.

template<typename T >

Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::unit_vector_free

(

const Eigen::Matrix< T, Eigen::Dynamic, 1 > &

x

)

Transformation of a unit length vector to a "free" vector However, we are just fixing the unidentified radius to 1.

Thus, the transformation is just the identity

Template Parameters

T	type of elements in the vector

Parameters

x	unit vector of dimension K

Returns

Unit vector of dimension K considered "free"

Definition at line 22 of file unit_vector_free.hpp.

void stan::math::validate_non_negative_index ( const char *  var_name, const char *  expr, int  val  )

inline

Definition at line 12 of file validate_non_negative_index.hpp.

template<typename T >

T stan::math::value_of ( const fvar< T > &  v )

inline

Return the value of the specified variable.

Parameters

v

Variable.

Returns

Value of variable.

Definition at line 17 of file value_of.hpp.

template<typename T >

std::vector<typename child_type<T>::type> stan::math::value_of ( const std::vector< T > &  x )

inline

Convert a std::vector of type T to a std::vector of child_type<T>::type.

Template Parameters

T	Scalar type in std::vector

Parameters

[in]

x

std::vector to be converted

Returns

std::vector of values

Definition at line 21 of file value_of.hpp.

double stan::math::value_of ( const var &  v )

inline

Return the value of the specified variable.

This function is used internally by auto-dif functions along with value_of(T x) to extract the double value of either a scalar or an auto-dif variable. This function will be called when the argument is a var even if the function is not referred to by namespace because of argument-dependent lookup.

Parameters

v

Variable.

Returns

Value of variable.

Definition at line 23 of file value_of.hpp.

template<typename T >

double stan::math::value_of ( const T  x )

inline

Return the value of the specified scalar argument converted to a double value.

See the primitive_value function to extract values without casting to double.

This function is meant to cover the primitive types. For types requiring pass-by-reference, this template function should be specialized.

Template Parameters

T	type of scalar.

Parameters

x	scalar to convert to double

Returns

value of scalar cast to double

Definition at line 24 of file value_of.hpp.

template<typename T , int R, int C>

Eigen::Matrix<typename child_type<T>::type, R, C> stan::math::value_of ( const Eigen::Matrix< T, R, C > &  M )

inline

Convert a matrix of type T to a matrix of doubles.

T must implement value_of. See test/math/fwd/mat/fun/value_of.cpp for fvar and var usage.

Template Parameters

T	type of elements in the matrix
R	number of rows in the matrix, can be Eigen::Dynamic
C	number of columns in the matrix, can be Eigen::Dynamic

Parameters

[in]

M

Matrix to be converted

Returns

Matrix of values

Definition at line 25 of file value_of.hpp.

const std::vector<double>& stan::math::value_of ( const std::vector< double > &  x )

inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters

x	Specified std::vector.

Returns

Specified std::vector.

Definition at line 42 of file value_of.hpp.

template<int R, int C>

const Eigen::Matrix<double, R, C>& stan::math::value_of ( const Eigen::Matrix< double, R, C > &  x )

inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Template Parameters

R	number of rows in the matrix, can be Eigen::Dynamic
C	number of columns in the matrix, can be Eigen::Dynamic

Parameters

x	Specified matrix.

Returns

Specified matrix.

Definition at line 51 of file value_of.hpp.

int stan::math::value_of ( int  x )

inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters

x

value

Returns

input value

Definition at line 55 of file value_of.hpp.

const std::vector<int>& stan::math::value_of ( const std::vector< int > &  x )

inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters

x	Specified std::vector.

Returns

Specified std::vector.

Definition at line 57 of file value_of.hpp.

template<int R, int C>

const Eigen::Matrix<int, R, C>& stan::math::value_of ( const Eigen::Matrix< int, R, C > &  x )

inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Template Parameters

R	number of rows in the matrix, can be Eigen::Dynamic
C	number of columns in the matrix, can be Eigen::Dynamic

Parameters

x	Specified matrix.

Returns

Specified matrix.

Definition at line 71 of file value_of.hpp.

template<>

double stan::math::value_of< double > ( double  x )

inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters

x

value

Returns

input value

Definition at line 40 of file value_of.hpp.

double stan::math::value_of_rec ( const var &  v )

inline

Return the value of the specified variable.

Parameters

v

Variable.

Returns

Value of variable.

Definition at line 17 of file value_of_rec.hpp.

template<typename T >

double stan::math::value_of_rec ( const fvar< T > &  v )

inline

Return the value of the specified variable.

T must implement value_of_rec.

Template Parameters

T	Scalar type

Parameters

v

Variable.

Returns

Value of variable.

Definition at line 22 of file value_of_rec.hpp.

template<typename T >

std::vector<double> stan::math::value_of_rec ( const std::vector< T > &  x )

inline

Convert a std::vector of type T to a std::vector of doubles.

T must implement value_of_rec. See test/math/fwd/mat/fun/value_of_rec.cpp for fvar and var usage.

Template Parameters

T	Scalar type in std::vector

Parameters

[in]

x

std::vector to be converted

Returns

std::vector of values

Definition at line 23 of file value_of_rec.hpp.

template<typename T , int R, int C>

Eigen::Matrix<double, R, C> stan::math::value_of_rec ( const Eigen::Matrix< T, R, C > &  M )

inline

Convert a matrix of type T to a matrix of doubles.

T must implement value_of_rec. See test/unit/math/fwd/mat/fun/value_of_test.cpp for fvar and var usage.

Template Parameters

T	type of elements in the matrix
R	number of rows in the matrix, can be Eigen::Dynamic
C	number of columns in the matrix, can be Eigen::Dynamic

Parameters

[in]

M

Matrix to be converted

Returns

Matrix of values

Definition at line 24 of file value_of_rec.hpp.

template<typename T >

double stan::math::value_of_rec ( const T  x )

inline

Return the value of the specified scalar argument converted to a double value.

See the primitive_value function to extract values without casting to double.

This function is meant to cover the primitive types. For types requiring pass-by-reference, this template function should be specialized.

Template Parameters

T	Type of scalar.

Parameters

x	Scalar to convert to double.

Returns

Value of scalar cast to a double.

Definition at line 24 of file value_of_rec.hpp.

const std::vector<double>& stan::math::value_of_rec ( const std::vector< double > &  x )

inline

Return the specified argument.

See value_of_rec(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters

x	Specified std::vector.

Returns

Specified std::vector.

Definition at line 43 of file value_of_rec.hpp.

template<int R, int C>

const Eigen::Matrix<double, R, C>& stan::math::value_of_rec ( const Eigen::Matrix< double, R, C > &  x )

inline

Return the specified argument.

See value_of_rec(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Template Parameters

R	number of rows in the matrix, can be Eigen::Dynamic
C	number of columns in the matrix, can be Eigen::Dynamic

Parameters

x	Specified matrix.

Returns

Specified matrix.

Definition at line 50 of file value_of_rec.hpp.

template<>

double stan::math::value_of_rec< double > ( double  x )

inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters

x	Specified value.

Returns

Specified value.

Definition at line 40 of file value_of_rec.hpp.

template<typename T >

return_type_t<T> stan::math::variance ( const std::vector< T > &  v )

inline

Returns the sample variance (divide by length - 1) of the coefficients in the specified standard vector.

Template Parameters

T	type of elements in the vector

Parameters

v	specified vector

Returns

sample variance of vector

Exceptions

<code>std::invalid_argument</code>

if the vector has size zero

Definition at line 22 of file variance.hpp.

var stan::math::variance ( const std::vector< var > &  v )

inline

Return the sample variance of the specified standard vector.

Raise domain error if size is not greater than zero.

Parameters

[in]

v

a vector

Returns

sample variance of specified vector

Definition at line 42 of file variance.hpp.

template<typename T , int R, int C>

return_type_t<T> stan::math::variance ( const Eigen::Matrix< T, R, C > &  m )

inline

Returns the sample variance (divide by length - 1) of the coefficients in the specified matrix.

Template Parameters

T	type of elements in the vector
R	number of rows in the matrix, can be Eigen::Dynamic
C	number of columns in the matrix, can be Eigen::Dynamic

Parameters

m

matrix

Returns

sample variance of coefficients

Exceptions

<code>std::invalid_argument</code>

if the matrix has size zero

Definition at line 49 of file variance.hpp.

template<int R, int C>

var stan::math::variance

(

const Eigen::Matrix< var, R, C > &

m

)

Definition at line 61 of file variance.hpp.

template<typename T >

const std::vector<T>& stan::math::vec_concat ( const std::vector< T > &  v1 )

inline

Ends the recursion to extract the event stack.

Parameters

v1	Events on the OpenCL event stack.

Returns

returns input vector.

Definition at line 17 of file vec_concat.hpp.

template<typename T , typename... Args>

const std::vector<T> stan::math::vec_concat ( const std::vector< T > &  v1, const Args...  args  )

inline

Gets the event stack from a vector of events and other arguments.

Parameters

v1	A event stack to roll up.
args	variadic arcs passed down to the next recursion.

Template Parameters

Args	Types for variadic.

Returns

Vector of OpenCL events

Definition at line 29 of file vec_concat.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::von_mises_lpdf	(	T_y const &	y,
		T_loc const &	mu,
		T_scale const &	kappa
	)

Definition at line 17 of file von_mises_lpdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >

return_type_t<T_y, T_loc, T_scale> stan::math::von_mises_lpdf ( T_y const &  y, T_loc const &  mu, T_scale const &  kappa  )

inline

Definition at line 109 of file von_mises_lpdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >

return_type_t<T_y, T_shape, T_scale> stan::math::weibull_lpdf ( const T_y &  y, const T_shape &  alpha, const T_scale &  sigma  )

inline

Definition at line 140 of file weibull_lpdf.hpp.

template<typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >

return_type_t<T_y, T_alpha, T_tau, T_beta, T_delta> stan::math::wiener_lpdf ( const T_y &  y, const T_alpha &  alpha, const T_tau &  tau, const T_beta &  beta, const T_delta &  delta  )

inline

Definition at line 197 of file wiener_lpdf.hpp.

template<typename T_y , typename T_dof , typename T_scale >

return_type_t<T_y, T_dof, T_scale> stan::math::wishart_lpdf ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  W, const T_dof &  nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &  S  )

inline

Definition at line 96 of file wishart_lpdf.hpp.

template<class RNG >

Eigen::MatrixXd stan::math::wishart_rng ( double  nu, const Eigen::MatrixXd &  S, RNG &  rng  )

inline

Definition at line 15 of file wishart_rng.hpp.

Variable Documentation

const double stan::math::CONSTRAINT_TOLERANCE = 1E-8

The tolerance for checking arithmetic bounds In rank and in simplexes.

The default value is 1E-8.

Definition at line 12 of file constraint_tolerance.hpp.

const double stan::math::E = boost::math::constants::e<double>()

The base of the natural logarithm, \( e \).

Definition at line 17 of file constants.hpp.

const double stan::math::EPSILON = std::numeric_limits<double>::epsilon()

Smallest positive value.

Definition at line 61 of file constants.hpp.

const double stan::math::INFTY = std::numeric_limits<double>::infinity()

Positive infinity.

Definition at line 46 of file constants.hpp.

const double stan::math::INV_SQRT_PI = inv(SQRT_PI)

Definition at line 132 of file constants.hpp.

const double stan::math::INV_SQRT_TWO = inv(SQRT_TWO)

The value of 1 over the square root of 2, \( 1 / \sqrt{2} \).

Definition at line 29 of file constants.hpp.

const double stan::math::INV_SQRT_TWO_PI = inv(SQRT_TWO_PI)

Definition at line 144 of file constants.hpp.

const double stan::math::LOG_EPSILON = std::log(EPSILON)

Definition at line 170 of file constants.hpp.

const double stan::math::LOG_HALF = std::log(0.5)

Definition at line 154 of file constants.hpp.

const double stan::math::LOG_PI = std::log(pi())

Definition at line 146 of file constants.hpp.

const double stan::math::LOG_PI_OVER_FOUR = LOG_PI / 4.0

Definition at line 148 of file constants.hpp.

const double stan::math::LOG_SQRT_PI = std::log(SQRT_PI)

Definition at line 150 of file constants.hpp.

const double stan::math::LOG_TEN = std::log(10.0)

The natural logarithm of 10, \( \log 10 \).

Definition at line 41 of file constants.hpp.

const double stan::math::LOG_TWO = std::log(2.0)

The natural logarithm of 2, \( \log 2 \).

Definition at line 35 of file constants.hpp.

const double stan::math::LOG_TWO_PI = LOG_TWO + LOG_PI

Definition at line 166 of file constants.hpp.

const double stan::math::LOG_ZERO = std::log(0.0)

Definition at line 152 of file constants.hpp.

const std::string stan::math::MAJOR_VERSION = STAN_STRING(STAN_MATH_MAJOR)

Major version number for Stan math library.

Definition at line 22 of file version.hpp.

const std::string stan::math::MINOR_VERSION = STAN_STRING(STAN_MATH_MINOR)

Minor version number for Stan math library.

Definition at line 25 of file version.hpp.

const double stan::math::NEG_LOG_PI = -LOG_PI

Definition at line 160 of file constants.hpp.

const double stan::math::NEG_LOG_SQRT_PI = -LOG_SQRT_PI

Definition at line 162 of file constants.hpp.

const double stan::math::NEG_LOG_SQRT_TWO_PI = -std::log(SQRT_TWO_PI)

Definition at line 158 of file constants.hpp.

const double stan::math::NEG_LOG_TWO = -LOG_TWO

Definition at line 156 of file constants.hpp.

const double stan::math::NEG_LOG_TWO_OVER_TWO = -LOG_TWO / 2.0

Definition at line 164 of file constants.hpp.

const double stan::math::NEG_LOG_TWO_PI = -LOG_TWO_PI

Definition at line 168 of file constants.hpp.

const double stan::math::NEG_TWO_OVER_SQRT_PI = -TWO_OVER_SQRT_PI

Definition at line 136 of file constants.hpp.

const double stan::math::NEGATIVE_EPSILON = -EPSILON

Largest negative value (i.e., smallest absolute value).

Definition at line 66 of file constants.hpp.

const double stan::math::NEGATIVE_INFTY = -INFTY

Negative infinity.

Definition at line 51 of file constants.hpp.

const double stan::math::NOT_A_NUMBER = std::numeric_limits<double>::quiet_NaN()

(Quiet) not-a-number value.

Definition at line 56 of file constants.hpp.

opencl_context stan::math::opencl_context

static

Definition at line 383 of file opencl_context.hpp.

const std::string stan::math::PATCH_VERSION = STAN_STRING(STAN_MATH_PATCH)

Patch version for Stan math library.

Definition at line 28 of file version.hpp.

const double stan::math::POISSON_MAX_RATE = std::pow(2.0, 30)

Largest rate parameter allowed in Poisson RNG.

Definition at line 71 of file constants.hpp.

const double stan::math::SQRT_PI = std::sqrt(pi())

Definition at line 130 of file constants.hpp.

const double stan::math::SQRT_TWO = std::sqrt(2.0)

The value of the square root of 2, \( \sqrt{2} \).

Definition at line 23 of file constants.hpp.

const double stan::math::SQRT_TWO_OVER_SQRT_PI = SQRT_TWO / SQRT_PI

Definition at line 138 of file constants.hpp.

const double stan::math::SQRT_TWO_PI = std::sqrt(TWO_PI)

Definition at line 142 of file constants.hpp.

const double stan::math::TWO_OVER_SQRT_PI = 2.0 / SQRT_PI

Definition at line 134 of file constants.hpp.

const double stan::math::TWO_PI = 2.0 * pi()

Definition at line 140 of file constants.hpp.